Pets are unlockable creatures that provide passive bonuses during gameplay. They don't have any impact on your combat abilities, but they will make it faster to progress.
In order to unlock each Pet, you'll need to repeatedly farm certain stages and defeat 5000 of the corresponding enemy. The Difficulty tier doesn't matter, and your progress continues across each one.
Here's some of the abilities you can expect.
- Increased Common Chest Drop Chance Multiplier
- Increased Stage Boss Drop Chance Multiplier
- Increased Gold Per Kill
- Increased Exp Gain
As you can see, every bonus makes farming just a little bit easier, and since they are permanent effects, you'll want to unlock them as soon as possible.
Additionally, no matter which Pet you decide to deploy, you will still receive the benefits of them all. So feel free to pick your favorite, or mix it up to match your mood.