The order Anguilliformes, is a relatively large group that contains about 800 species. You might be suprised to know, however, that not everything that we call an eel is actually in this family. In fact, many of the species we call eels are distantly related. For example, what we call electric eels, spiny eels, wolf eels, bobtail snipe eels and gulper eels are not true eels. Almost all the eels in this family are marine. The most notable exception are the freshwater eels in the family Anguillidae. These eels are catadromous, meaning they live their adult lives in freshwater streams and then migrate back to the ocean to breed. These freshwater eels make monumental migrations of thousands of miles, all back to one small area of the ocean, just south of Bermuda. These migrations, are equivalent, although sometimes the opposite of, the migrations of Salmon who migrate up freshwater streams to breed.
The 800 species in this order are divided between the following 15 families. Note that some sites will classify eels into 19 families. The most recent taxonomy however, puts these 4 extra families into the order Saccopharyngiformes  (gulper eels, pelican eels, bob-tail snipe eels)
To learn more about the eels, we recommend the following resources.