You need two plastic tire levers, a pump, and either a repair kit or a new inner tube. Let all air out of the tube and push the tire towards the center, away from the rim, all around the wheel. Insert the spoon-shaped part of one tire lever between the rim and the tire at the place where you suspect the puncture, scoop the tire out of the rim, and hook the other end into a spoke. Very carefully insert the spoon end of the lever only as far as needed to prevent pinching the tube. Repeat two spoke holes away from the first with the other lever. Repeat until you can pull the tube out of the tire.
If you are on the road and have a spare tube, it is usually best to install it and fix the punctured tube at home, in a warm and dry place. This requires removing the wheel by releasing the quick-release lever in the axle. In the case of the rear wheel, shift to the smallest rear sprocket first. Mountainbikes sometimes require unscrewing the quick-release nut partway because of ridges in the dropout that prevent accidentally losing a wheel when the quick-release comes loose.
Small holes can be fixed by glueing a patch. I have never managed to reliably patch a hole more than a millimeter or two in length. If the hole is large, an air bubble will form under the patch and find a way out. Snakebites require a large patch that covers both holes. Pump up the tube to find the hole. Let nearly all the air out of the tube again, clean and dry the tube around the hole, and use the sandpaper to roughen the area. File off seam protrusions. Apply a thin coat of glue that is larger than the patch. Patiently wait until it dries, which takes a couple of minutes (impatience is rewarded with a patch that comes loose soon). Apply the patch and press on hard at the center and all around the edges. Continental patches seem to be the best. Always push the air out of the glue tube when replacing the cap so it won't dry up quickly.
To reinstall the tube, inflate it so it just barely holds its shape. Insert the valve and push the tube into the tire there. Working away from the valve, push the tube into the tire while slightly pushing it towards the valve. Alternatingly work from the left and right of the valve to prevent the valve from being pulled in one direction. If the tube is so long that a loop remains after it is fully installed, don't squeeze it in but pull it all the way out again and retry, this time pushing it harder towards the valve. The goal is to keep the tension of the tube constant along the circumference of the wheel. It's easier than it sounds. If you have a tire liner, always keep it centered, and place the overlap of the liner ends at the side opposite from the valve.
Push the open end of the tire into the rim beginning at the valve, taking special care that the tube is not trapped between rim and tire, and work away from the valve. Usually it is hard to push in the last bit on the opposite side. First try pushing the tire towards the center of the rim while pulling it towards the remaining loop. If this is not sufficient, let all air out of the tire.
If all this fails (and only then), you need the tire levers again. Put a little bit of air into the tube so it holds its round shape and insert the spoon-shaped end of a tire lever between tire and rim, but with the "wrong" side up - not hooking into the rim but with the inner side of the curve towards the tire. This makes it harder to push the tire in but reduces the risk of pinching the tube. Begin near the ends and work towards the middle of the section not yet pushed in. You can use the hook end of the other lever to keep the tire from pushing out of the rim at the other end.
Sometimes the wall of the tire gets punctured. This is dangerous because the tube will form a bubble there that can explode with a loud bang. Torn tire walls are unfixable but you can keep the tube inside by inserting a folded dollar bill or some other denomination of your choice. Paper money is incredibly strong and will last years if the hole is not too large. If the threads in the side wall are damaged so much that the tire bends out of shape after inflating, it's unfixable, you need a new tire. Obviously, don't do dangerous downhills with such a fix, it will probably fail at the worst possible moment.
See also an article on "Fixing a Rubbing Wheel"
