Be Considerate. A lot of people love pets. Others don't, and some are actually scared of them for a variety of different reasons. It's important to be respectful of this. If you are walking your pet and some neighbours want to play with it, great! But don't expect everyone to feel that way. You wouldn't want to make someone who is afraid uncomfortable, anxious, or frightened.
Be Respectful. If you have a dog, take it out to do its business and don't let it happen in or near the building. Accidents happen, and if they do ensure you clean it up immediately. If you need a little help, speak with your building's caretaker or property manager.
Keep a leash on at the right time. If your apartment has a courtyard or outdoor area, it is more acceptable to let your pet run without a leash. However, common (and indoor) areas, such as the lobby, are places where the leash should be kept on. Some buildings even require that the owner carry their pet through the lobby.
Take necessary precautions. This is more relevant to dogs because they are regularly walked. If you know your dog bites, use preventative measures such as a muzzle or wait for an empty elevator. All residents should feel safe in the apartment and you wouldn't want anyone to get hurt!
Elevator manners. Speaking of elevators, be mindful when others are in the elevator with you. Shorten the leash or even ask if your neighbour is okay with being in the elevator with your pet as a courtesy.
Mind the weather. If it's raining, snowy, or slushy, it would be nice to wipe off your pet's paws before re-entering the building!