The electric eel is not a true eel.  Nor is the "wolf eel" in the picture below. The moray eel, on the other hand, is a true eel.  What's the difference?  The easy answer is that a true eel is a member of the fish order Anguilliformes, and that other eel-like fish are not.  But this begs the question:  what characteristics are used to place a fish in the eel order?  

Moray Eel

Wolf Eel

1.  They have a dorsal fin that is continuous with the caudal and anal fin.
2.  They have very small , embedded scales or no scales.
3.  As juveniles and adults, members of the family Anguillidae live in fresh water, but they breed in the ocean, where most other members of the eel order, including the Moray, spend their entire lives.  These freshwater eels are skin-breathers, absorbing oxygen into dense beds of capillaries that lie very close under the skin.  
4.  They are "slippery as an eel" due to the production of slime.  In the green moray, yellow slime overlying a blue body produces its greenish color.    
5.  Larval eels are flattened ribbonlike creatures and look so different from the adults that some larval forms were once considered to be separate species of fish!
6.  They have numerous sharp teeth.  
7.  Freshwater eels are an important food fish in many countries.  Moray eels, on the other hand, are toxic.  

The "wolf eel" is a member of the Family Anarhichadidae (wolf fishes). More information about wolf eels can be found here. Electric "eels" are members of the Family Electrophoridae in the Order Gymnotiformes or knifefish.   An interesting thing about this order is that their closest relatives are in the catfish order, the Siluriformes.   What are some differences between electric eels and  true eels? 

Electric Eel

1.  The electric eel is a freshwater fish, and lays its eggs in fresh water, while most true eels are marine for at least part of their lives, and all true eels spawn in the ocean.  
2.  Electric eels have no teeth.  
3.  Electric eels have no dorsal fin.  
4.  Electric eels are air-breathers.  In the warm, oxygen-poor waters of the Amazon basin, breathing air is an important survival strategy.