Photo by Qurren
There are generally two main reasons why a transmission would slip:
1. The transmission's components may be getting worn. This could range from worn or broken transmission bands to worn clutch plates. 
2. There is a problem with the transmission fluid. Either the fluid levels are too low, there is inadequate fluid pressure, or the fluid itself is burnt.
A good place to start is by checking your transmission fluid using the dipstick in the engine bay. This dipstick should be easy to identify since it will generally either be labeled with the letters "TRANS" or will be red. 
If you check your transmission fluid and find that it is low, there is a high likelihood that you have a leak. Before adding more transmission fluid, you should try to fix the leak. There are liquid products on the market that can be added to the transmission fluid to help seal any leaks. This can be a quick, affordable way to deal with the problem. Otherwise, you may have to take your car to the shop to have them check for leaks.
If you check the transmission fluid and the levels are fine, you are most likely dealing with a worn part. In this case, you will probably have to take your car to the shop to have it repaired unless you have the tools and experience required to work on your transmission yourself.
You can see how easy it is to use the one of the sealing fluids to stop leaks here: