-Brake before going into the turn, using both brakes.

If you have a lot of traction:
-Push the outside foot down and lean to the inside (if you have traction).
-Enter the corner wide, hit the apex with the bike near the inside edge 
 and leave the corner wide.
-Do not use the front brake if you are turning at the bike's limit.  The 
 front tire is using all its traction for turning.  If you use your front 
 brake, it will lose its grip and wash out.  A front wheel slide is almost 
 impossible to recover.  A back end slide is easier to recover.  Also, the 
 brake tire is doing less work than the front, therefore, you can use some 
 of its "spare" traction for braking.

If you are turning on loose surfaces, keep this in mind:
-This technique involves keeping the bike relatively upright; instead, the
 body is leaned in the direction of the turn.
-Transfer weight slightly forward.  Push down on the outside pedal.
-Twist your upper body to face the trail.  Align your upper body so 
 that your upper body is slightly leaning toward the inside of the turn. 
-Push down on handlebar on the outside and pull up on the inside.

Others have pointed out:
Dave Blake [dblake@eureka.wbme.jhu.edu]
In loose stuff, steering is definitely the preferred way to turn.  This
is the reason many roadies with good bike skills cannot handle
tight singletrack very well. 

To steer, put your weight on the inside of the turn.  Turn your front wheel 
toward the turn, and hold your bike upright.  Even if one or both of your
wheels begin to skid you can easily recover.  In contrast, if you lean hard
through a turn on loose material and either wheel loses traction, you will 
be picking gravel out of your leg.

You almost always want your weight centered between your wheels.
This means you move your butt further back as the terrain gets
steeper.  Learn to feather your front brake.  Let off on the brake
when your wheel hits an obstacle, and hit it harder when you have
a smooth even braking surface.  Many people do not learn to feather the
brake, so they put their weight too far towards the rear to keep from
endoing.  This rear weight shift results in too little weight being placed
on the front wheel, so that you cannot easily steer.

lrtredwa@rdyne.rockwell.com
I find that the most consistent mistake that I make when turning on 
downhills is to shift too much weight to the rear.  This causes the 
front wheel to become too light causing it to wash out :-(  If my 
weight is more evenly distributed on the bike, I find that I am also 
in a position to recover if the front starts to wash out (if I'm not 
going too fast) although it is not often I can reover from a front wheel 
washout.  

John Stevenson [johnstev@world.net]
Look at the inside of the turn, not the outside.  Your body tends to
subconsciously point in the same direction as your eyes, so this keeps you
focused on staying tight in the curve, not straying to the outer edge.

Blaine Bauer [bbauer@cisco.com]
One thing that I've learned through hard knocks is sharp turning -
especially in loose soil. We have some trails that constantly wind through
the woods, and have little room on each side (re: trees).

I've found that negotiating sharp turns at some reasonable speed is easier
when the seat is an inch or so lower than normal. The trick is to lean the
bike (but not the rider). This is really just a variation of normal turning.

- Point the inside leg in the direction of the turn (knee away from the
frame), putting all weight on the outside pedal.
- Push down on the inside handlebar. At this point almost all weight 
should be distributed between the inside grip and the outside pedal. This 
is much easier with a rigid fork - with a suspension fork, you really 
have to bear down on the handlebar (a grunt may be required!).
- At this point the bike is leaning under the rider, with the seat 
anywhere from under the thigh to just under the knee. The rider's 
weight is centered over the point where the two wheels are in contact with 
the ground, so there isn't a  washout problem even in loose conditions.

This method will feel very uncomfortable at first. Pushing the handlebars
away from oneself is...well, disquieting. The best way to practice this is
to do figure-8 turns in a driveway. When you've got it down, hose down the
driveway and then try it. If you can make sharp turns on wet concrete you
can do it in loose soil.