Pablo is the first cat I've ever owned and he's been kind enough to teach me all about cat behavior and anatomy through the years. For instance, it was when I trimmed his claws for the first time that I learned cat's claws have an outer layer, or sheath (I call them shells). They shed these sheaths periodically, at which point new razor sharp claws are revealed. When cats scratch at a scratching post (or furniture!) they're not actually doing it to sharpen their claws but to drop the sheaths on them in order to free the new claws underneath. This is why it's essential that a cat have something appropriate to scratch otherwise he will direct his energy towards furniture or carpet. Occasionally I'll see one of the cats tugging at their claws with their mouth in an attempt to remove a shell.
We trim the cat's claws every 2-3 weeks to save our furniture from utter ruin and Andy's adorable face from being scratched to shreds. I use a dedicated pair of human cuticle trimmers to do the job - they look like small nail clippers with a slanted top. I like these clippers because they were cheap, they're small and allow for precision trimming, and they've remained sharp for 7 years running. They're just the right size for cat's claws but I wouldn't recommend them for dogs. You can find them in the nail supply aisle of just about any big box store if you're interested.