How do you fix the sprocket on a mountain bike?

Answer 1

Sounds like the freewheel (the clutch prom). You need a special tool to remove it in order to be able to replace it with a new one you can find the tool at a shop near you most of all have them

Answer 2

There isn't much in a rear sprocket that can reasonably be fixed. Most common action is simply to replace it. You need the correct freewheel puller to get it off though. www.parktool.com has a page about repairs and tools.

If your problem is that its sticking or not engaging you can try to flush and relube it. Put bike on side, flush plenty of WD-40 or similar through gap in freewheel, see if you get any gunk out. Then dribble in some runny oil(engine oil will do) and see if it helped.