If you have a cat, when you bring your puppy home it is important you properly introduce him or her to your cat:
Place your puppy on the floor. Do not leave them alone to get to know each other as that could be dangerous for both puppy and cat.
Put your puppy into a crate for a short time as this will allow the cat to eye him up and down at a safe distance without the cat feeling threatened.
If the puppy is used to a lead, try placing a lead on your puppy and walking him around the room while your cat is in the room, making sure your cat has a high place it can retreat to if your puppy decides to chase the cat. Otherwise, hold the puppy on your knee so they can see each other but cannot get to each other yet.
Once you place the puppy on the floor, if your puppy is fixating his gaze on your cat call him by his new name. Once you have his attention give him plenty of praise and a treat.
Do this routine a few times until your puppy and your cat are used to the idea.
Eventually both animals will relax and give in to the idea both of them having to share their home.
A baby gate or stair gate will help your cat to have some peace of mind so he or she can roam knowing your puppy cannot follow upstairs. Do not spoil things by leaving them alone together whilst you go out until you can be certain they are more comfortable.