Do you own a dog, a cat or something more exotic and caged, like a parrot or iguana? That's fine, and it's probably fine with your landlord, but it's something that will have to be agreed upon with very specific terms. Some apartments don't allow any pets at all, or they allow caged pets, cats instead of dogs (as cats usually aren't as destructive as dogs), or cats and dogs under a certain size. Ask your landlord before you move in for the details on pets in your apartment -- it's easier to find a place that will let you bring your pet than trying to renegotiate your lease after you've already moved in.
If you're allowed to have an animal, you'll most likely be charged some kind of fee or deposit for the right. This could be either a move-out cleaning fee or a "pet deposit" charge, similar to the security deposit, from which any post-residential pet damage costs will be deducted. Another option is that the pet deposit fee is completely non-returnable.
Pet rules may be subject to negotiation. Say, for example, the landlord has a 30-pound (13.6-kilogram) weight limit on pets, and your dog weights 35 pounds (15.9 kilograms). That just means they don't want large, potentially messy dogs -- you can probably compromise on that. And if you plan to get a pet later on, tell your landlord, as you may need to pay a fee upon move-in or renegotiate the lease upon the animal's arrival.
If there isn't anything in the lease agreement laying out pet rules or possible fees, ask your landlord, and get any important information in writing. Whatever you do, don't try to sneak in a pet without telling your landlord. Sure, you'll avoid paying the deposit, but if you get caught, that's a violation of the lease terms (as you claimed on the lease to not keep pets) which could lead to eviction.
What if you need to move out, but your lease isn't up yet? It's not the end of the world.