Repairing damage to the sidewall of a tire is not as easy as fixing damage to the trod area, but it can be done. If the tire is sliced, it should be discarded and replaced immediately. However, if it is simply punctured (by a nail, screw, or small stick), there are cheap and easy kits that can be purchased at any auto supply store that can restore the integrity of the tire.
Purchase a tire plug kit from an auto parts store. It only runs a couple of dollars and can be extremely useful in a pinch. In fact, most tire shops use one for their own repairs. There is no need to get a big fancy kit with a thousand plugs; just get one with a few plugs and a tube of cement (included in the kit).
Remove the puncture culprit, if it is still present. It is easiest to use the plug kit when the tire is inflated, but it is not completely necessary. When you remove the object, the tire will begin to rapidly lose air, so some speed in applying the plug is necessary.
Place a small amount of cement on the plug installer (it looks kind of like a drill bit with a T-handle). Then, jam it into the puncture. Move it in and out and turn it around as well, as if you were pumping up a bike tire. The idea is to apply the cement while cleaning and shaving the puncture hole at the same time.
Place a small amount of cement on the centre of the plug itself (they kind of look like round beef jerky sticks). Then, insert it into the puncture with the same T-handled tool. The plug should be stretched across the puncture and then the middle should be inserted. Depending on the size of the hole, it can difficult to shove in, so be careful not to push it all the way through. The end result should be the plug folded in half with both ends of it sticking out toward you. Cut the ends of the plug down and the tire is sealed. Remember to fill it back up with air before use!
It is a good idea to wear gloves because the cement is quite sticky!
Test the tire out at low speeds to ensure the plug is going to stay in before hitting the highway.