Do different types of Poké Balls exist in the anime?

First off, we'll ignore unusual, villain-only Poké Balls like Mewtwo's "Clone Balls" or the Iron-Masked Marauder's Dark Balls because they don't appear in the games and are only used in one episode or movie. They almost always come with some sort of unethical quality (e.g. the Dark Balls make a Pokémon evil) and only exist to establish how threatening the villain can be.

In the anime, a number of Pokéballs just have aesthetic improvements, like diamonds or symbols, but are functionally the same. The first Poké Ball with a different name that we see are the Safari Balls in Episode 35 (an episode that did not get translated into English, but was called "?????????? The Legend of Miniryu [Dratini]" in Japanese). In all honesty, these seem to be just branded Poké Balls used only in the Safari Zone. Nothing is functionally special about them.

The next is the GS Ball, which was originally going to contain Celebi. Then when it was decided that the fourth Pokémon movie was going to feature it, the plotline was dropped as redundant and Ash forgot about it after giving it to Kurt. It was never used to capture a Pokémon.

The Generation 2 Apricorn balls, created by Kurt, were featured in the anime and had all the variants they did in the game (Love Ball, Heavy Ball, Lure Ball, etc).

The other specialty balls (e.g. the ones introduced in Generation 3) have never been mentioned and have only occasionally appeared as cameos, notably the Japanese credits song for Diamond and Pearl "Which One ~ Is It?".

In the movie Pokémon 4Ever - Celebi: Voice of the Forest, Ash time travels into the past and meets Sammy, who has an old Poké Ball that requires one to unscrew the top to let a Pokémon out. Unfortunately, we don't see it capture a Pokémon, so we don't know how effective it is.

The Master Ball, which in the games can catch anything, made its only appearance in the Advanced Generation Episode 75 "Whiscash and Ash". A fisherman named Sullivan has been going on a Moby Dick-style hunt to catch a Whiscash that he has nicknamed "Nero". When he finally corners it, Sullivan draws out a Master Ball, which shocks everyone. He tosses it at Nero and it swallows it whole and then runs away. Yup. A Master Ball, which can capture any Pokémon, failed to capture a Pokémon because it swallowed it! Sullivan was quite understandably angry.

The Great Ball and Ultra Ball finally appear in XY Episode 79 "A Frenzied Factory Fiasco!". Despite being in Generation 1, it took 878 episodes for a Great Ball and Ultra Ball to appear in the anime! The heroes go to a Poké Ball factory and see them being created on the conveyor belt along with most of the other non-Apricorn variant Poké Balls. However, none of them are used except for a Luxury Ball. Meowth almost gets captured in one, stating that it felt really nice inside.

So to recap, the only Poké Balls that we have seen used are:

• Regular Poké Balls
• Safari Balls, which appear to be functionally identical to Poké Balls
• The Apricorn Balls, which had their special effects detailed
• The Master Ball, which failed to capture a Pokémon
• The Luxury Ball, which feels very nice for the Pokémon inside

Although the other Pokéballs exist, they appear to be much rarer and we have never actually seen them used, so it's not clear how much better they are.

It's likely that the reason that regular Pokéballs are almost exclusively used, rather than these specialty balls, is because it's more difficult for the animators to be consistent with the Pokémon coming out of special Pokéballs. Indeed, quite a few episodes have animation errors where Pokémon caught in Generation 2 Apricorn balls (e.g. Ash's Totodile) are released from regular Pokéballs. In the early days, there was a lot of reused footage as well and if you have Pokémon coming out of different types of balls, then you can't reuse your footage. It seems the animation team just decided that it's easier to have all Pokémon come out of the same type of Pokéball.