• Billywig
• Bowtruckle
• Demiguise
• Diricawl
• Doxy
• Dung Beetle
• Erumpent
• Fwooper
• Glow Bug
• Graphorns
• Grindylow
• Marmite
• Mooncalf
• Murtlap
• Niffler
• Nundu
• Obscurus aka Obscurial
• Occamy
• Swooping Evil
• Thunderbird

• Billywig

  Several times throughout the film we see a blue beetle flying through the city called a Billywig.

  The Billywig is an insect native to Australia. It is around half an inch long and a vivid sapphire blue, although its speed is such that it is rarely noticed by Muggles and often not by wizards until they have been stung. The Billywig’s wings are attached to the top of its head and are rotated very fast so that it spins as it flies. At the bottom of the body is a long thin sting. Those who have been stung by a Billywig suffer giddiness followed by levitation. Generations of young Australian witches and wizards have attempted to catch Billywigs and provoke them into stinging in order to enjoy these side effects, though too many stings may cause the victim to hover uncontrollably for days on end, and where there is a severe allergic reaction, permanent floating may ensue. Dried Billywig stings are used in several potions and are believed to be a component in the popular sweet Fizzing Whizbees.
  ^{_{—Fantastic Beasts and Where to Find Them by J.K. Rowling, Pottermore (link)}}

• Bowtruckle

  This green little guy who follows Newt around is a Bowtruckle.

  The Bowtruckle, which eats insects, is a peaceable and intensely shy creature but if the tree in which it lives is threatened, it has been known to leap down upon the woodcutter or tree-surgeon attempting to harm its home and gouge at their eyes with its long, sharp fingers. An offering of woodlice will placate the Bowtruckle long enough to let a witch or wizard remove wand-wood from its tree.
  ^{_{—Fantastic Beasts and Where to Find Them by J.K. Rowling, Pottermore (link)}}

• Demiguise

  

  The Demiguise is found in the Far East, though only with great difficulty, for this beast is able to make itself invisible when threatened and can be seen only by wizards skilled in its capture.
  The Demiguise is a peaceful herbivorous beast, something like a graceful ape in appearance, with large, black, doleful eyes more often than not hidden by its hair. The whole body is covered with long, fine, silky, silvery hair. Demiguise pelts are highly valued as the hair may be spun into Invisibility Cloaks.
  ^{_{—Fantastic Beasts and Where to Find Them by J.K. Rowling, Pottermore (link)}}

• Diricawl

  The Diricawl _{^{[thanks to Anu7 for reminding me of this one]}}

  The Diricawl originated in Mauritius. A plump-bodied, fluffyfeathered, flightless bird, the Diricawl is remarkable for its method of escaping danger. It can vanish in a puff of feathers and reappear elsewhere (the phoenix shares this ability; see page 32). Interestingly, Muggles were once fully aware of the existence of the Diricawl, though they knew it by the name of “dodo.” Unaware that the Diricawl could vanish at will, Muggles believe they have hunted the species to extinction. As this seems to have raised Muggle awareness of the dangers of slaying their fellow creatures indiscriminately, the International Confederation of Wizards has never deemed it appropriate that the Muggles should be made aware of the continued existence of the Diricawl.
  ^{_{—Fantastic Beasts and Where to Find Them by J.K. Rowling, [no Pottermore extract available, this is directly from the source (screenshot)]}}

• Doxy

  One of the flying creatures is the Doxy, reminiscent of the pixie

  The Doxy is often mistaken for a fairy (see page 16) though it is a quite separate species. Like the fairy, it has a minute human form, though in the Doxy’s case this is covered in thick black hair and has an extra pair of arms and legs. The Doxy’s wings are thick, curved, and shiny, much like a beetle’s. Doxies are found throughout northern Europe and America, preferring cold climates.
  ^{_{—Fantastic Beasts and Where to Find Them by J.K. Rowling, [no Pottermore extract available, this is directly from the source (screenshot)]}}

• Dung Beetle

  Giant Dung Beetles are often in the background rolling balls of dung

  Dazed, he swats them out of the way. Behind him a large dung beetle rolls a giant ball of dung.
  ^{_{—Fantastic Beasts and Where to Find Them: The Original Screenplay by J.K. Rowling, SCENE 47, INT. Newt's Case, Animal Area (screenshot)}}

  Dung beetles exist in the real world too, but these are magical at least in their giant size.

• Erumpent

  The beast that gets lost in Central Park is an Erumpent.

  The Erumpent is a large grey African beast of great power. Weighing up to a tonne, the Erumpent may be mistaken for a rhinoceros at a distance. It has a thick hide that repels most charms and curses, a large, sharp horn upon its nose and a long, rope- like tail. Erumpents give birth to only one calf at a time.
  ^{_{—Fantastic Beasts and Where to Find Them by J.K. Rowling, Pottermore (link)}}

• Fwooper

  The brightly coloured Fwooper makes an appearance

  A tiny bright pink bird, the Fwooper, flies past and comes to rest on a little perch, suspended from midair.
  ^{_{—Fantastic Beasts and Where to Find Them: The Original Screenplay by J.K. Rowling, SCENE 47, INT. Newt's Case, Animal Area (screenshot)}}

  The Fwooper is an African bird with extremely vivid plumage; Fwoopers may be orange, pink, lime green, or yellow. The Fwooper has long been a provider of fancy quills and also lays brilliantly patterned eggs. Though at first enjoyable, Fwooper song will eventually drive the listener to insanity8 and the Fwooper is consequently sold with a Silencing Charm upon it, which will need monthly reinforcement. Fwooper owners require licences, as the creatures must be handled responsibly.
  ^{_{—Fantastic Beasts and Where to Find Them by J.K. Rowling, [no Pottermore extract available, this is directly from the source (screenshot)]}}

• Glow Bug

  Among the floating bubbles we see the mysterious Glow Bugs

  They swarm upward, mingling with Doxys, Glow Bugs, and Grindylows, which float through the air.
  ^{_{—Fantastic Beasts and Where to Find Them: The Original Screenplay by J.K. Rowling, SCENE 47, INT. Newt's Case, Animal Area (screenshot)}}

• Graphorns

  The giant beasts we see inside Newt's cavernous suitcase are Graphorns.

  The Graphorn is found in mountainous European regions. Large and greyish purple with a humped back, the Graphorn has two very long, sharp horns, walks on large, four-thumbed feet, and has an extremely aggressive nature. Mountain trolls can occasionally be seen mounted on Graphorns, though the latter do not seem to take kindly to attempts to tame them and it is more common to see a troll covered in Graphorn scars. Powdered Graphorn horn is used in many potions, though it is immensely expensive owing to the difficulty in collecting it. Graphorn hide is even tougher than a dragon’s and repels most spells.
  ^{_{—Fantastic Beasts and Where to Find Them by J.K. Rowling, Pottermore (link)}}

• Grindylow

  We see a Grindylow swimming in a floating bubble

  They swarm upward, mingling with Doxys, Glow Bugs, and Grindylows, which float through the air.
  ^{_{—Fantastic Beasts and Where to Find Them: The Original Screenplay by J.K. Rowling, SCENE 47, INT. Newt's Case, Animal Area (screenshot)}}

  A horned, pale-green water demon, the Grindylow is found in lakes throughout Britain and Ireland. It feeds on small fish and is aggressive towards wizards and Muggles alike, though merpeople have been known to domesticate it. The Grindylow has very long fingers, which, though they exert a powerful grip, are easy to break.
  ^{_{—Fantastic Beasts and Where to Find Them by J.K. Rowling, [no Pottermore extract available, this is directly from the source (screenshot)]}}

• Marmite

  Newt is seen bottle-feeding a luminescent creature with tentacles, which the VFX house behind the shot named a Marmite^{[thanks to Valorum for identifying this one]}

  ANGLE ON NEWT, now cradling a luminescent creature with sprouting alien-like tendrils. He feeds the creature with a bottle, while carefully watching how Jacob handles the Mooncalves—he recognizes a kindred spirit.
  ^{_{—Fantastic Beasts and Where to Find Them: The Original Screenplay by J.K. Rowling, SCENE 47, INT. Newt's Case, Animal Area (screenshot)}}

  This creature isn't named or referenced in the screenplay or the original textbook, but has been identified as a Marmite by consulting the VFX studio that created the effect. All the credit for this identification goes to Valorum, who provided the answer to the question here:

  On the creatures side, Method handled the shot of Newt bottle-feeding a young marmite – a tentacled cross between a dust mite and squid with a transparent body. Production provided live action plates with the actor, and Method artists added the CG creature to the shot, digitally wrapping 10-foot long tentacles around Newt and creating an underwater feel.

  _{^{DELUXE’S METHOD STUDIOS CONTRIBUTES VFX MAGIC TO WB’S “FANTASTIC BEASTS AND WHERE TO FIND THEM}}

• Mooncalf

  The big-eyed creatures that jump excitedly at Jacob are the Mooncalves

  The Mooncalf is an intensely shy creature that emerges from its burrow only at the full moon. Its body is smooth and pale grey, it has bulging round eyes on top of its head, and four spindly legs with enormous flat feet. Mooncalves perform complicated dances on their hind legs in isolated areas in the moonlight.These are believed to be a prelude to mating (and often leave intricate geometric patterns behind in wheat fields, to the great puzzlement of Muggles).
  Watching Mooncalves dance by moonlight is a fascinating experience and often profitable, for if their silvery dung is collected before the sun rises and spread upon magical herb and flower beds, the plants will grow very fast and become extremely strong. Mooncalves are found worldwide.
  ^{_{—Fantastic Beasts and Where to Find Them by J.K. Rowling, [no Pottermore extract available, this is directly from the source (screenshot)]}}

• Murtlap

  The Murtlap is the bald, rat-like porcupine that attacked Jacob in his apartment.

  The Murtlap is a ratlike creature found in coastal areas of Britain. It has a growth upon its back resembling a sea anemone. When pickled and eaten, these Murtlap growths promote resistance to curses and jinxes, though an overdose may cause unsightly purple ear hair. Murtlaps eat crustaceans and the feet of anyone foolish enough to step on them.
  ^{_{—Fantastic Beasts and Where to Find Them by J.K. Rowling, [no Pottermore extract available, this is directly from the source (screenshot)]}}

• Niffler

  The first critter we're introduced to is the Niffler.

  The Niffler is a British beast. Fluffy, black and long-snouted, this burrowing creature has a predilection for anything glittery. Nifflers are often kept by goblins to burrow deep into the earth for treasure. Though the Niffler is gentle and even affectionate, it can be destructive to belongings and should never be kept in a house. Nifflers live in lairs up to twenty feet below the surface and produce six to eight young in a litter.
  ^{_{—Fantastic Beasts and Where to Find Them by J.K. Rowling, Pottermore (link)}}

• Nundu

  The Nundu, spotted among the creatures in Newt's possession, looks like a mix between a tiger and a pufferfish.

  The camera pans up to reveal another magnificent creature, the Nundu—looking almost exactly like a lion, it has a large mane that bursts forth when it roars. It stands proudly on a large rock, roaring at the moon. Newt scatters food at its feet and purposefully it moves on.
  ^{_{—Fantastic Beasts and Where to Find Them: The Original Screenplay by J.K. Rowling, SCENE 47, INT. Newt's Case, Animal Area (screenshot)}}

  This east African beast is arguably the most dangerous in the world. A gigantic leopard that moves silently despite its size and whose breath causes disease virulent enough to eliminate entire villages, it has never yet been subdued by fewer than a hundred skilled wizards working together.
  ^{_{—Fantastic Beasts and Where to Find Them by J.K. Rowling, Pottermore (link)}}

• Obscurus aka Obscurial

  ^{_{Note: this is the Obscurus from inside the suitcase Note: this Obscurial is Credence and does not come from inside Newt's suitcase}}

  We push inward, toward a small oleaginous black mass suspended in midair—an Obscurus.
  Jacob, intrigued, moves into the snowscape to get a closer look.

  The mass continues to swirl, emitting a disturbed, restless energy. Jacob reaches out to touch it.

  ...

  JACOB
  What the hell is this thing?

  NEWT
  It's an Obscurus.
  ^{_{—Fantastic Beasts and Where to Find Them: The Original Screenplay by J.K. Rowling, SCENE 47, INT. Newt's Case, Animal Area (screenshot)}}

  JACOB
  (sotto voce, to Tina)
  Can someone please tell me what this Obscurial—Obscurius thing is? Please?

  TINA
  (also sotto voce)
  There hasn’t been one for centuries—

  NEWT
  I met one in Sudan three months ago. There used to be more of them but they still exist. Before wizards went underground, when we were still being hunted by Muggles, young wizards and witches sometimes tried to suppress their magic to avoid persecution. Instead of learning to harness or to control their powers, they developed what was called an Obscurus.

  TINA
  (off Jacob’s confusion)
  It’s an unstable, uncontrollable Dark force that busts out and—and attacks . . . and then vanishes . . .
  ^{_{—Fantastic Beasts and Where to Find Them: The Original Screenplay by J.K. Rowling, SCENE 61, INT. MACUSA Cell (screenshot)}}

  An Obscurus (plural Obscuri) is the creature, an Obscurial is the person hosting it (e.g. Credence is an Obscurial, Dumbledore's sister is probably an Obscurial etc. the creature itself, in the bubble, is an Obscurus).

• Occamy

  The Occamy is a serpent that Newt finds hiding out in Macy's.

  The Occamy is found in the Far East and India. A plumed, two-legged winged creature with a serpentine body, the Occamy may reach a length of fifteen feet. It feeds mainly on rats and birds, though has been known to carry off monkeys. The Occamy is aggressive to all who approach it, particularly in defence of its eggs, whose shells are made of the purest, softest silver.
  ^{_{—Fantastic Beasts and Where to Find Them by J.K. Rowling, Pottermore (link)}}

• Swooping Evil

  When it probes an Auror's ears, Newt tells the Swooping Evil to "LEAVE HIS BRAINS!"

  A large, butterfly-like creature that emerges from a small object, possibly a cocoon.
  Appearance: Blue and green winged beast
  ^{_{—Swooping Evil, Pottermore (link)}}

• Thunderbird

  The enormous bird Newt lets fly free in New York is called a Thunderbird.

  The Thunderbird is a magical American bird closely related to the phoenix
  ^{_{—History of Magic in North America, 1920s Wizarding America by J.K. Rowling, Pottermore (link)}}

Note: Not all the creatures in the suitcase, featured in the movie Fantastic Beasts and Where to Find Them (2016), are described or even mentioned in the Fantastic Beasts & Where to Find Them (2001) fictional textbook, published by J.K Rowling. The Obscurus only gets mentioned as a publisher (not the creature), and the Marmite, Nundus, Swooping Evil, Thunderbird, Glow Bugs and Dung Beetles don't even get mentioned. Other canonical sources like Pottermore and Fantastic Beasts and Where to Find Them: The Original Screenplay (by J.K. Rowling) were used to identify and describe the creatures in the film. Please consult and inspect the source links I added individually for each citation if the exact source concerns you.