-Shift before you hit the climb.  The only way to know which gear is best 
 for your terrain is from practice.  It is very hard on your drivetrain if 
 you shift in the middle of your climb.
-Seated is better for long distance and/or loose conditions.  Standing
 is good for hammering up a short steep section with good traction.
-If you find you are in too easy a gear, upshift once in the back.  Do 
 not dump a bunch of gear at once.

Seated Climbing:
-If you are going to stay seated, move slightly forward on the saddle.
-Move your head close to the stem to keep the front from coming up.
-Don't pull up on the handlebar, instead, pull backward with every stroke.  
-Keep your body relaxed, and shoulders square to the trail.
-Put the bike in a low gear and spin.

Standing Climbing:
-If you decide to stand up, put the bike in a higher gear.  You can't 
 spin as fast, but you can apply more power per stroke.
-Crouch down so that your butt is right in front of the saddle.  Your 
 elbows should be bent and the chest should be just above the stem.

-For both methods, try to look for the smoothest line and look for slight 
 dips on the climb.  These will offer you a great opportunity to rest for 
 a bit.

Others added:
Bill Rod [smts!brod@msss.attmail.com]           
I agree in general, but IMHO I would recommend using the middle or big ring 
when standing.  I've found standing while using the granny gear causes 
overtorqueing (sic) and hence wheelspin. 
Medek@aol.com
On longer climbs, alternate the position (standing, sitting) for a short
period of time.  Each position uses a different set of muscles and altering
the position will give you an opportunity to rest different muscle groups.

John Stevenson [johnstev@world.net]
Tim Gould's maxim always seems relevant here: "Start easy, finish hard".  In
other words, start a climb in the very lowest gear you have, and shift up
as you get comfortable.  That way you can gauge your fitness and the
severity of the slope, rather than getting commited to trying to stomp up a
1km 20 per cent grade in 36/28.

Long climbs, particularly at high altitude, are places where a stupidly low
gear will come in useful.  I'm talking 20/28 *or lower* here.  Here's the
scenario: you're happily plodding uphill in, say, the 22/28 that is now a
typical low gear on a Shimano equipped bike.  You come to a slightly steeper
technical section that requires an increase in your effort level.  You power
over the problem, sending your heart hammering into the upper end of your
anaerobic range.  What you could really do with now is an even lower gear to
allow you to recover, but the idiots who spec most off-the-peg bikes don't
seem to realise this=8A IMNAAHO 20/28 is the maximum sensible bottom gear 
for a mountain bike that is used in real mountains, and I know people who
have gone to the current technical limit, 20/32.

Brian Adams [adams@cs.unr.edu]
-Pull your elbows in on very steep, slow climbs.  It helps to keep your front
 wheel from wandering.

Tom Hewitt [hewitt@crayalb.cray.com]
  I'm 44years old, and while slower than most riders, can usually clean hills
that younger riders more fit don't. In my case the key for climbing really
nasty long technical hills, is to practice going as slow as practical on those
sections of lessor technical difficulty. This conserves energy for the
difficult sections, where all-out effort is required. In addition
balance in an extreme climbing situation is different from balance in
a level ground situation, and can only be learned by spending lots of time
fighting to keep your balance.

Rik Allen [richard@prl.research.philips.com]
-Standing is better on very technical/slimey climbs. You can move your
weight around much more to hop wheels over obstacles that they would bog
down on otherwise. Bunnyhopping sideways out of ruts is almost
impossible seated. Plan ahead. Keep your body moving smoothly up the
hill, and make the bike move under you.
-On climbs with obstacles (wet tree routes) these gears cause their own
problems. Seriously low gears require too many pedal revs to get over
whatever is in your way, and you end up moving so slowly that balance
becomes harder, with more risk of wheelspin or flipping over backwards.
IMHO, anyway. Less than 24 inches becomes a problem for me getting over
tree roots.