Dumping used kitty litter into a drain is a bad idea. Because it's heavier than water, it sinks to the lowest point of the drain, which is the bottom of the P-trap. The main ingredient in kitty litter is an absorbent substance called Fuller's earth, and as the litter settles, this ingredient forms a clay-like, solid mass resistant to most drain chemicals. Removing kitty litter with a plunger or auger is impractical -- you need to disassemble the trap to clear it out and get the drain working again. Clearing it from the toilet often involves pulling the toilet.
Plunge the affected drain to create enough of an opening to allow standing water to drain. Cover the sink overflow holes with tape or rags to prevent a break in suction that renders the plunger ineffective.
Put a bucket under the P-trap to catch water and unscrew the trap from the sink tailpiece. Let water drain, then unscrew the other end of the trap from the drain. If you can't unscrew the P-trap nuts by hand, use adjustable pliers to loosen them.
Invert the trap over the bucket and empty all the water. Take the trap outside and and scrape out all the kitty litter from inside using a screwdriver. Spray water from a garden hose into the trap as you're cleaning it to flush out loose particles of clay and debris.
Unscrew the aerator from your faucet and screw on a pipe thread to hose thread converter, which is an inexpensive plumbing fitting you can find at any hardware store.
Screw the hose to the converter, turn on the cold water and spray water into the drain. If the water backs up, there is probably an obstruction at a bend in the pipe. Insert an auger (standard drain snake) and work it through the obstruction until the drain runs freely.
Re-connect the P-trap by hand-tightening the nuts you unscrewed to remove the trap. When they feel tight enough, test the trap by filling the sink with water, opening the stopper and letting the water drain. If either connection leaks, tighten it with adjustable pliers.