How can I loosen a tight sink handle?
Assuming this is a faucet that fits variable widths, each tap has screws underneath the sink which allow you to adjust the height of the tap relative to the sink.  The exact technique used to adjust the height varies, but generally in my experience you have to:


Loosen the locking screws/nuts - these lock the larger nut in
place (if you have any) 
Adjust the height of the faucet with the large nut under the sink 
Tighten the locking screws/nuts again


As for why this could happen - changes in ambient temperature could have caused a slight adjustment in the height of the faucet above the sink - so the bottom of the tap could be turning against the sink where it didn't before.  (This happened to me with both my hot and cold taps at different times.)

Here's an example of what they look like - these are my own faucets.  Each tap has a brass ring holding it in place right under the sink, and two locking screws that lock the brass ring in place.  The height of the tap is set by turning the brass ring.