Climbing

Climbing, or riding uphill, can be quite a balancing act if the terrain starts to get steep. On gentle inclines, there is really no difference in technique from level riding except that you have to pedal harder and keep your weight centered on the bike with a slight forward lean. Select a comfortable gear and try to stay seated in the saddle. Make sure that you saddle height is high enough for good leg extension. Anticipate the coming terrain, and be prepared for roots, rocks, or logs that may require some momentum to carry you over. Avoid a big burst of power, and avoid expending too much energy for any one obstacle if the uphill trail is long or extended.
Pace yourself, establish a nice breathing rhythm, and maybe use your bar end extenders to change your hand position and to stretch your body out a bit and shift your weight a bit forward. A glance at the top (or as far as you can see) is ok, but stay focused on the trail in front of you and take it in sections.

For the steeper climbs, you must expend a sizeable amount of energy just to make it up short sections & be prepared to really do some work. Long, steep climbs are a tremendous aerobic challenge and your muscles can load up quick with lactic acid. Many of these climbs may be beyond your skill and strength level - you may have to be satisfied with only doing a section and then walking the rest. In addition to the high energy expenditure, you really need to be in the proper gear and have the correct fore and aft balance on the bike.
Try to ride in a gear higher than you feel you may need - don't automatically select the lowest gear unless the terrain is steep enough to require it. Try to anticipate the uphill sections, and get into the proper gear for climbing as soon as you can before you get on the steep parts. It is sometimes hard to downshift if you are under load pedaling uphill and need to go to a lower gear. Even for steeper climbs try to stay seated in the saddle, crouch down, and keep your nose forward, and your weight forward. If your rear tire starts to loose traction, just lean back a bit. Avoid standing to pedal unless you are very near the top of the climb.
Use your handlebar ends or "cowhorns" to help keep your weight forward. Practice pedaling smooth and strong and avoid quick downward pedaling. Sit on the nose of the saddle and keep most of your weight on the pedals and not the saddle. Your upper body and torso may need to be almost parallel to the ground in a deep crouch position on the steepest terrain.

If you can see that it is too steep to continue riding, be prepared to clamp on BOTH brakes so that when you stop to dismount the bike does not roll back. Dismount on your favorite side (mine is the left) and tilt the bike sideways a little to the left when dismounting so that you can swing your leg over for an easy dismount. Sometimes, a dismount on a really steep uphill section can be a bit tricky so be careful. If you are ridng in cleats, they may be a bit slippery and lose traction on the steepest terrain, so walking your bike up the hill may be difficult in some circumstances. Take note of the terrain and tricky spots you may have to encounter on your descent, and give a little thought to your downhill line in the harder sections you may encounter on the ride back.

Steep Climbing

This is really just the opposite of way steep descending (DUH!). You need to find the balance point as you climb, which moves forwards on the bike the steeper you go. On a fairly steep climb leaning your body forward (Road bike aero style) puts enough of your weight forward on the bike to do the trick. On the steeper pitches you have to resort to moving forward on the saddle. If you think about descending for a moment, you end up stretched out, as you move back, whilst still holding on to the bar (well most of us keep holding on to the bar) When climbing you do the opposite and end up scrunched up towards the bar. This feels (and looks) a bit strange, but when the trail is really steep maintaining this position (and keeping pedaling) are the keys to going up.
Most people say 'hover just above the nose of the saddle'. Well this is true, but if you have a WTB saddle (or any with a long nose) it is possible to sit on the saddle nose and keep pedaling. This really works but can be hugely uncomfortable so hovering it is. Use a very low gear (DUH again) and spin the pedals furiously whilst keeping your body still. This requires enormous energy from such a cramped position.
There is a way of getting even more power and traction from your rear wheel however. When climbing in the weight forward (crunched up) position, there's a limit to how much power you can apply to the pedals without the tyre slipping. On a long steep climb, smooth pedalling from the balance position is essential to prevent wheel spin, which occurs on the down push (the most powerful part of the pedalling stroke).
For very short steep pitches pulling up hard on the bars with your hands, to counteract the force of pushing on the pedal, can be incredibly effective. What happens is that if you pull up with your left hand whilst pushing down with your left leg, the rear wheel is momentarily forced down in to the ground. If your efforts are synchronised the wheel won't spin out even with your strongest powered pushes. This kind of power is essential if you want to climb the 'impossible' steep pitch on your favourite rides.
The technique needs to be practised (although some people get it right first try) and is not dissimilar to the roadie 'honking' from a forward position. Remember to use left hand to pull up when pushing with left leg, and right hand to pull up when pushing with right leg. Don't forget to lean forward, move your weight forward and pedal at a quick rhythm, but don't hit the slope too hard or you may lose balance.

Other essential climbing tips:
With all of these the crunched forward position is the key:

Keep up a fast rhythm of smooth pedaling.
Hop the front wheel up over steps / roots.
On a hard-tail you may need to hop up the rear end too.
On a full suspension it's often better to sit down on the saddle to maintain rear wheel traction over roots.
Practice front and rear hops by riding up a kerb without front or rear wheel touching it.
At the crest of a short climb throwing your body forward often gives that little extra oomph needed to get over the top.