Here's an easy rule to remember: Never give your pet any over-the-counter medication without clearing it with your veterinarian first.
That's a good rule to remember in general, but in particular, it applies to painkillers. Although you can safely give aspirin to arthritic dogs, the smaller size and different metabolism of cats make aspirin a dangerous proposition for them. Acetaminophen, the active ingredient in Tylenol, can kill your cat, as can some of the newer, longer-lasting painkillers available in nonprescription form for human use.
If your pet is in pain, call your veterinarian immediately. Cats in particular are very stoic, and if you're noticing your pet's discomfort, he's really suffering and needs immediate care. As for chronic pain, your veterinarian can prescribe something that's effective and pet-safe.