Ron Sawicki is a professional mountain bike mechanic (he is also Pau Sawicki's trainer). Here he gives some insightful tips on replacing your inner tube whilst out in the trails plus I added a bit on actually repairing the puncture itself. Often done back home.

(courtesy of pctv1)

• Make sure you take with you the following: 3 Tyre levers, replacement tube, puncture repair kit, small pump,
• If the puncture is in your back wheel then a useful tip is to get your gears into the smallest cog. Makes it much easier to get the wheel off.
• Release remaining air and unseat the tyre with the levers (one is normally enough)
• Take out the tube and check inside of tyre slowly with hand for sharp thorns, stones etc.
• Blow up your replacement tube slightly  and insert in tyre first. Make sit much easier to put on and helps stops the inner tube getting a pinch flat when being put back on the wheel rim.
• Put on rim and seat in the try with your hand (sometimes you may need to use a lever gently to put the tyre back on)
• Blow up tyre ride off.

Fixing the Puncture itself

• As above clearly to get the tube out.
• Locate the hole by dunking the tube into water and remove any thorns, etc. from inside the tyre. Clean and dry the tube, and then apply a film of rubber solution.
• Really important to wait a minute until it dries. Then when nearly dry, remove the silver foil from the patch and smooth it into place. Then peel off the cover
• Lots of patches are self adhesive these days.
• Then as above to get it back on