A manual flash only fires at a fixed power and it is up to you to adjust the aperture, shutter speed and ISO of your camera so that the scene is correctly exposed. There is usually some sort of chart on the flash to help you to set the appropriate aperture, etc.
An auto flash has a sensor that can determine when enough light has been fired and can stop its own output. It usually has a few settings to allow for various ranges of aperture. Most auto flashes can also function as a manual flash, and the more advanced ones can also fire at 1/2, 1/4, 1/8, ... power.
You would usually want to buy an auto flash if you're a beginner because it is not easy or convenient to use manual flash. Plus an auto flash can usually be used as a manual one, but not vice versa.