It says in the summoning spell that when you summon an animal it is a fey fiend or celestial (your choice). What does this do? Do they get any abilities or buffs from these like types?
It only matters due to other game features
Firstly there are no innate bonuses from creature type. There are however some features which care about creature type:
Firstly, the familiar is not a beast anymore, so spells like beast bond do not affect it.
A few (cleric) spells care about creature type and about celestial, fey, and fiend in particular. Hallow prevents any of them from entering the area. Temple of the Gods and Forbiddance prevent a caster's choice of those creatures (+Elementals and undead) from entering the area (or deals 5d10 damage per turn without a save, which is fairly equivalent).
If you are a Circle of the Shepherd druid a few of your features care about beasts or fey you've created or summoned, giving it buffs like additional hit points. Thus choosing the specific type is relevant if you are such a druid with access to find familiar. (See this Q&A of mine for more details.)
There are numerous other game features, like Ranger's Favoured Enemy, that could hypothetically be relevant, but which are even less likely to come up.
These are all fairly niche interactions, so you are pretty much free to choose whichever you like, or feel fit your character, irrespective of mechanical benefits.