The type of hose repair needed depends, of course, on the type of leak you have. Here are two different possibilities for a leak, and how to fix them.
Ordinary garden hoses tend to leak at one of two points, either at the fitting or at a kink.
Kinks are usually the result of trying to get by with a cheap hose. They have non-existent or ineffective inner linings. The mesh lining is what keeps the hose from folding over on itself, instead forcing a smooth curve. It's the sharp bends that produce stress on the tube, leading to breaks. A few dollars of prevention is worth a huge headache of cure here. Buy quality.
For breaks that do occur somewhere along the length, there are two choices.
If the crack is small, you may get along (for a time, at least) by simple taping. Special plastic tape is available, but standard black plastic electrical insulating tape works well, too. Make sure the area around the leak is dry. The adhesive on tape dissolves readily in water. Simply wrap in a tight spiral well above and below the leak.
For larger breaks, slice the hose with an ultra-sharp X-acto style knife on either side of the break. Make the edges clean, dry and cut perpendicular to the axis of the hose. An imaginary line running down the center of a straight hose.
Buy a hose repair kit.
They come in two styles - for end fittings or for interior repairs. Those are broken down into two sub-types: male and female. Male fittings insert into the hose opening, female fittings wrap on the outside of a hose or another fitting. A male inserts into a female.
Some simply insert into the hose, some have threads. The threads aren't expected to actually screw into the hose, but they can help create a better seal. Wrap the latter type with several winds of plumbers tape to ensure a good seal.
Warm the hose in the sun or near a fire (but don't apply a butane lighter directly) to expand the hose. This allows male fittings to be inserted into the hose more easily. Once inserted, allow to cool and you'll have snug fit.
For end-hose fitting (the part a water nozzle or faucet screws onto), the procedure is essentially the same. The same procedure is required if cracks appear where the (usually brass, sometimes plastic) fitting meets the hose, or if the fitting has become broken or out of round.
Slice off the fitting, clean the end and make sure the hose surface is perpendicular to the hose axis. Warm the hose for a male fitting, cool it for a female fitting (the majority). Before pressing on or inserting the fitting, test it on the faucet or nozzle to ensure a snug fit.
Insert or clamp as above.