There are various forms of mountain bike racing, and one is sure to be right for you.

Cross Country (XC)
This is the most common type of racing, and the best for beginners. The terrain varies depending on the region, but it usually features dirt roads, singletrack, hill climbs, rocks, roots, mud, and more.

Beginner races will be about 1 hour in length and usually avoid the extremely technical trails.

Short-Track XC
Short track XC racing usually takes place on a fairly mild cross country course, usually dirt roads or double-track. The course is short, fast, fairly smooth, and twisty.

These races are based on time. For example, you may race for 20 minutes then do two more laps. [Laps may be .5 to 1.5 miles)

Downhill (DH)
Downhill racing involves racing down a hillside against the clock. You want to beat other racers' times.

This can be quite dangerous, and it usually requires an expensive full-suspension bike. It's not the most beginner friendly racing, but it doesn't involve lots of hill climbs!

Marathon/Endurance
This is basically an XC race that goes for 40-100 miles. They also come as "12 hour" or "24 hour" races where you do as many laps as possible within a certain time period.

Mountain Cross
Like a combination of downhill and short track, these races put 4 racers against each other on a (mostly) downhill course.

There are a few places on and offline that can help you find a race. It can be kind of hard to find a race depending on your area, but this should help.

First, go to local bike shops and see if anyone there can recommend a local race.

Then you might want to search the internet for "your town + mountain bike race"

After that, you could try typing your town's name in at a site like BikeReg.com and other event calendars like at bicycling.com.

Race websites are listed below: