Electric eels generate current through their highly specialized nervous system. This interacts with three electricity generating organs; the main organ, the Sach's organ, and the Hunter's organ. The nervous system of electric eels is able to connect with these organs and has control over whether electricity is released or not. The collection and release of energy is facilitated by the electrolytes (in this case, salts in water), which connect the organs and allow for the flow of current. This produces an electric charge, which can be release via neural control of the eel. 

When the eel has identified its prey, it can signal the electric cells in the electricity generating organs, opening up ion channels and depolarizing (reversing the polarity - the positive and negative charges on either side of the membrane) the cell membrane. This change in polarity generates a current. This current can be released into the environment. Different things can be done with the released energy. In the case of an electric eel in its natural habitat, it would use this mechanism to capture and kill prey. However, when you combine electric eels with the creativity of human, one can find that these eels can power human electrical equipment, such as a Christmas Tree.