Wikipedia will give you the most thorough set of answers to your question, but a short and simple answer may do just as well.
Just plain vanilla ".38" can refer to several different revolver cartridges. Some people use the term to describe the obsolete ".38 Colt" cartridge.
The ".38 Special" is a relatively long-case revolver cartridge, and is though to be a very basic, medium-power, self-defense load. Beginners usually find it easy to control and make good hits with.
The ".380," in the US, describes a low-power cartridge for semi-automatic pistols. It's about as weak as you can go, and still expect useful self-defense hits. However, it is usually used in small-size pistols which are, themselves, difficult to control and use well. (In British usage, ".380" could describe either a revolver or a semi-auto cartridge.)