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To start with, your repair kit requires: a spare tube, two tyre levers, glueless patches, a piece of high-density cloth (canvas = great, gaffa tape = acceptable) and a pump.

Firstly, remove the wheel. If it's the front, release the brake if you've got rim brakes or just undo the quick release skewer and unwind the nut of the QR if you've got discs.If it's the rear, change the gears so the chain is on the smallest rear cog, then release the brake/undo the QR skewer and remove the wheel. If you've got bolt-up hubs, you'd better have a set of allen keys on hand, but if you've got 15mm nuts, it's probably best just to fit thorn-resistant tubes and a sealing solution now as 15mm spanners are generally too bulky to carry.

Push one tyre lever between the bead and the rim, then lift the tyre over the edge. Insert lever 2 about 5 cms further around, again lifting the tyre, and repeat until you can pull the tyre off with your hands. Once the tyre's off, inspect it carefully inside and out for the cause of the puncture. If you can't find any sharp objects or cuts, inflate the old tube to find the cause. Two holes close together (like a snake bite) equals a pinch flat, whereas a hole on the inside edge of the tube could mean a problem with the rim - check it for protruding spokes or sharp edges.

Now to put the tyre back on! (Check the sidewall to see if there's an arrow.) Put one bead back on the rim, inflate the new tube a little, fit the valve into the rim, and start stuffing the tube evenly inside the tyre. If you've found a cut in the casing, use the cloth/tape to cover the hole from inside as you fit the tube. To fit the second bead, begin opposite the valve and push the bead over the rim with both hands, working evenly around to the valve until it starts to get tight. Now check that the bead is seated as low in the rim as possible (and that the tube isn't stuck between the bead and rim) by going along and rolling the tyre back and forth. Generally this step is enough to get the rest of the tyre onto the rim, but if you do need the levers, be extra careful not to pinch the tube! Now a critical step: push the valve stem up into the tyre and down again. This seats the valve properly. Failure to do this often causes the tyre to blow off the rim.

Finally, pump up the tyre (support the valve so you don't break it off with your pump), refit the wheel, connect your brakes if need be, and there you go!

Side note: glueless patches are for the times when you get another puncture on the same ride. They're not 100% reliable but will get you home. In this situation, remove the tube, inflate it, and rotate it close to your cheek to find where the air's escaping. When you've found the hole, polish the tube on your clothes to remove dust or dirt, whack the patch on, and finish as before.