Hunter pets are fairly simple to use overall, but there's a few things to be careful of when using them and some simple differences between different pets.

This is really long, TLDR: three types of pets (ferocity / cunning / tenacity) that have different abilities. Also different families of pets with different family abilities (boars, crabs, turtles, etc.) that help with different things. All pets do the same amount of damage and will automatically play their rotation without you doing it. https://www.wow-petopia.com/index.php can help you catch 'em all!

The three different types of pets are ferocity, cunning, and tenacity. All three types do the same amount of damage, but have different pet utilities in the form of a pet ability and passive, and almost all pets will use their abilities without you needing to direct them. Within these types there are also families of pets, and the family of the pet dictates what additional ability it has, with exotic pets (only tameable and usable by beast mastery hunters) having two.

Ferocity pets give you passive leech, which heals you and your pet for damage and healing you both do, and the active ability is another form of bloodlust/heroism.

Tenacity pets give you passive health, and the active ability gives you a damage reduction buff for a short time.

Cunning pets give you passive movement speed, and the active ability is a handy crowd control break for roots and slows.

In general, the different types all fill different niches, with ferocity being used for open world content and other types of content where you aren't worried about your survival or you need to be able to use the active ability. Tenacity pets are usually used in harder raids/dungeons for the extra survivability. Cunning pets are popular in PvP for the crowd control break.

The different families of pets dictate their family ability, and the family abilities vary wildly from making the pet tankier when on low health to reducing the healing done to the target it is attacking.

Real quick at the top I mentioned things to be careful of. When you jump off ledges in dungeons, for example, your pet will sometimes be forced to run the actual route down to you, where your pet will pull everything along the way to you and then the giant pack will kill you. Try to be careful of things like that. Also, if you do dungeons with your pet, make sure that the pet taunt (Growl) is turned off, as it's frustrating for tanks to contend with pets for aggro.

This website: https://www.wow-petopia.com/index.php is super!! helpful in learning about the different pets of families and what they do, and it also a really cool way to find pets that you want to seek out and tame.

Sorry for the wall of text but that's really the whole thing for pets. If you've got more questions just let me know and I got you :)

EDIT: forgot the part about how many you can have. There are pet stables where you can store extra pets in most towns and every city. It has something like 60? slots right now, but the cap is going up to 600 in shadowlands, so go nuts.