If you are not very handy, the best option is to replace it. They're pretty cheap, about $10, and generic versions are available.
Turn off the water valves under the sink, find the faucet-end of the sprayer line under there, use a wrench to undo it, and take it to your local hardware store to find a match. Install the new one, and turn on the water valves. Check for leaks under the sink.
If you are into taking things apart to see how they work, you could diagnose and possibly fix the sprayer. But it could be worn out or need parts that may not be available separately. Usually it's an o-ring or other seal gone bad, and the not-shutting-off thing is a weak spring. The water should be turned off again, and then the sprayer handle usually comes apart by removing a plastic plug over a screw on the sprayer tip, then the screw, and from there it's figure it out as you go.