Bathroom Sink Handles - What are they called?
Faucet handles or levers.  

If you don't know the brand and model it can be very tricky to track these down.  Perhaps the best thing to do is to take the lever piece off and go to a plumbing supply store and see if they can match it.  The problem I have run into is that I get something that works but not exactly.  So then you have to buy two levers and even then it isn't exact and you just spent $20-30 on something that isn't exact.  (I have also ran into this when I had model #)

I have had this issue many many times on flips and I simply just replace with a good working faucet.  Yours for instance I could pick up for $10 at a salvage store near me.  You could order something online for $30-50 that is brand new.