What is the basic power of Apocalypse?

The movie is bereft of details in Apocalypse's history. The closest it came to it CIA agent Moira Mactaggert's rough explanation of the cult's reasonings for worshiping Apocalypse.

Most of his powers and background are directly taken from his many comic and then cartoon appearances. Most likely due to licensing conflicts with Marvel, mentions of the background Celestial technology were cut out. But they can't be ignored as they are crucial to Apocalypse's backstory.

Quick Summary:

• Apocalypse was born a blue, deformed mutant.
• Originally had Super-Strength.
• Once stabbed to death, his full mutant powers developed, Healing and ability to grow his body.
• Stopped aging at adulthood.
• Everything else was enhancements from Celestial genetic manipulation and cyborg-like technology.
• This technology required him to change bodies due to it's overpowered state.
• Celestial Technology was referenced in the Concept stages of the movie, but unexplained in the film. Still provides background to character.

In 1996, an origin story was finally released a decade after his first appearance, explaining his original powers, and how he got more powerful. The "Rise of Apocalypse" 4 issue mini-series shows his birth. He is born a changeling, a type of mutant that has powers at birth. He's oddly shaped and blue skinned, as seen in the movie. Abandoned by his birth family, he's adopted by a tribe that believes in Survival of the Fittest, his entire POV and character motivation in the movie. As he ages, he has super-human strength. In issue 3, he's shown to float/rise with energy beams when thrown off a pyramid, which is kind of how he escapes his tomb in the movie. In issue 4, when the woman he loves rejects him, and someone stabs him, his body heals and grows in size, his basic power in any fight. In the movie, he gains his healing powers by transferring bodies, and only grows in size in the mental/astral world that he fights Xavier in. It's also revealed he doesn't age. His original form:

Finally, the mini-series also shows he had access to Celestial technology from almost the beginning. Long story short, Celestials are a god-like giant alien race, as old as the multiverse, with genetic manipulation technology. They created mutants by genetically altering early humans. In the MCU, Knowhere from Guardians of the Galaxy, is the head of a Celestial. The flashback to the destroyed planet is of another Celestial, Eson the Searcher. Thor and Asgardians are Gods of ants compared to Celestials.

The armor Apocalypse wears, and the transformation chamber, are Celestial technology. The technology he uses to transform his Horsemen in the comics is Celestial technology, while in the movie he just has the power.

Apocalypse has to hibernate to recharge, and has to change his body often because of the Celestial genetic changes are too powerful to withstand. In the movie, this is just used as a way to add new powers.

In the early concept stages for the movie, the director Bryan Singer posted production designer Grant Major's concept art that included "Ship", the Celestial Artificial Intelligence spaceship that provided Apocalypse all his extra powers.

In January, Singer also teased again, implying alien origins.:

"The way I describe him the most, the best, is he to me is the God of the Old Testament and all that comes with that. If there isn’t the order and the worship then I’ll open up the Earth and swallow you whole, and that was the God of the Old Testament. I started from there and when Oscar and I met we began discussing -- since he isn’t really God, he’s the first mutant perhaps, but he’s not God necessarily -- he’s imbued with certain unique powers. Some of them may or may not be from this Earth, we don’t know."

Considering the concept for the movie includes Celestial Technology, it's difficult to argue that the unexplained technology in the movie wasn't it, absent another explaination.

And the technology seen is very similar to the cartoon versions (pictured is X-Men Evolution, but its similar to the 90's The X-Men as well):

Thus, summing up his powers from the movie Apocalypse is shown an ancient mutant born with a variety of superhuman abilities who further augmented himself after merging with Celestial technology. The character has total control over the molecules of his body, enabling him to alter his form as it suits him, such as allowing his body to become extremely malleable and flexible or change his size, enhance his physical abilities, transform his limbs into weapons, wings, or jets, regenerate from fatal injuries, adapt his body to apparently any disease or hostile environment, as well as give himself virtually any superhuman power. He is also able to project and absorb energy, in addition to displaying capacities for telepathy and telekinesis. Apocalypse is also capable of technopathy, able to directly interface with the various technologies he has at his disposal. Thanks to the aid of his mutant abilities, special "regeneration" chambers, and changing bodies, Apocalypse has made himself effectively immortal.

From Wikipedia (of all places):

Apocalypse is an ancient mutant born with a variety of superhuman abilities who further augmented himself after merging with Celestial technology. The character has total control over the molecules of his body, enabling him to alter his form as it suits him, such as allowing his body to become extremely malleable and flexible or change his size, enhance his physical abilities, transform his limbs into weapons, wings, or jets, regenerate from fatal injuries, adapt his body to apparently any disease or hostile environment, as well as give himself virtually any superhuman power. He is also able to project and absorb energy, in addition to displaying capacities for telepathy and telekinesis. Apocalypse is also capable of technopathy, able to directly interface with the various technologies he has at his disposal. Thanks to the aid of his mutant abilities, special "regeneration" chambers, and changing bodies, Apocalypse has made himself effectively immortal.

So what… this guy's kind of like Mr. Fantastic on steroids? Yeah, his powers have always been sort of nebulous, but as long as he's cutting through X-Men teams like Kirstie Alley through Sizzler, I don't think the fans care.

-Frank Tieri, Interview about X-Men: Apocalypse vs Dracula

Aside from his superhuman powers, Apocalypse is extraordinarily intelligent, a scientific genius with knowledge in various areas of science and technology including physics, engineering, genetics and biology, all of which are more advanced than conventional science. Apocalypse has knowledge of Celestial technology that he uses for his own applications, such as altering mutants or humans. Apocalypse is also a skilled demagogue and a master strategist.

Apocalypse's blood can heal other mutants, but is fatal for humans. Apocalypse's blood can also restore his de-powered mutant descendants as is seen when a large dose of Apocalypse's blood regenerates the lost body part of Chamber and gave him a look similar to Apocalypse.