What type of faucet is this?
It's an installed faucet as far as I can make out.  To replace the stems it is not necessary to know the mfr and model (though might be helpful), instead just bring them in to a plumbing store (ideally) or Home Depot.

I once brought my broken version of this part:



in to a plumbing store.  Without out my saying a word, the counterhelp simply looked at it, said 'ok', walked back into the stock shelves and came back with the replacement kit.