There are a lot of different reasons for gutter leaks. The most common is because they are plugged up, in which case you need to clean them. Obviously they can get old and just need to be repaired.
There are three main things that go wrong with rain gutters to cause gutter leaks. These aren't the only things that can happen, just the most likely things that you will see.
Rain gutters can be weak at the seams. This is a good area to start looking if you have gutter leaks. You should get a ladder and walk around your house looking at all of the rain gutter seams. You will be looking for seams that are separating or may have holes on them from constant moving around in the wind.
As you find the seams that need to be fixed, you can go to the hardware store and get a tube of silicone sealant and a caulking gun, along with a razor blade. That should be all you need to fix the seams in your rain gutters.
After you do the repairs on the seams, you should allow 24 hours before water has to run through the rain gutters again.
Holes and Cracks in Your Rain Gutter
If you have a large area where there is no seems that is leaking, you will need to clean out the leaves and other debris that may be in the rain gutters and then let it dry out. Usually this type of leak means that your rain gutters have a rusted section in them.
You should sand the rust off with a wet or dry sand paper, about 180 grit. After you have done that you will need spray a rust protective paint or primer on the freshly sanded area of the rain gutter leaks.
Gutter leaks can be stopped by doing this. You will need to cut a piece of roofing flash or other metal about 1 inch longer then the hole. Lay down a coat of roofing cement and stick the roofing flash or other metal over the area that you have just cemented.
Allow 24 hours to dry after the repairs are complete.
If You have Copper Rain Gutters
In this case the repair techniques change and it will take a bit of skill to perform these repairs. The two solutions in the above repairs will no longer work when you are using copper rain gutters.
For small gutter leaks, you will need to re-solder the joints or apply a piece of copper flashing the size of the hole and solder it in to place. This will work for small leaks. Any big leaks will need to be fixed by a professional.
If you use the techniques above, you should be able to fix your gutter leaks without a problem and, unless you have copper rain gutters, it should be a fairly straight forward repair. In the end your gutter leaks will all be gone and things will be back to normal.