If your pet gets injured, your normal first-aid kit could suffice for most things. But it's a good idea to prepare a specific kit for your dog or cat so that you can be sure you will have everything you need in an emergency.
The American Pet Center outlines a comprehensive dog first-aid kit you can put together that includes a muzzle, nail trimmers, tweezers, a rectal thermometer, saline solution, an inflatable protective collar, bandages and a few other items. Cat first aid kits are similar; the only real difference is in the muzzle and protective collar.
Don't forget to include basic information in your kit as well, such as your veterinarian's contact details. A pet-specific first-aid manual might also be a good idea. You can also look up pet health situations at PetMD.
Making a Pet First Aid Kit [Animal Medical Center]