In an electrochemical cell, we physically separate the oxidation and reduction chemistry in different "compartments". The electrons from the oxidation are then run through an external circuit before being used in the reduction reaction. As this is moving negative charge from one location to another, we need to compensate for this by moving other charges to balance this displacement of charge. This is accomplished by using a salt bridge that allows the migration of spectator ions to balance the flow of electrons.
Each half of the electrochemical cell has an electrode to which the wire for our external circuit is connected. The chemistry takes place at the surface of this electrode. The electrode on the oxidation side is called the anode. The electrode on the reduction side is called the cathode. As a short hand, we simply refer to the two sides of the cell as the anode side and the cathode side.
For any given set of conditions, the chemistry is spontaneous in only one direction. When the cell is set up such that this is the direction we desire, we call the cell a voltaic cell (or a galvanic cell or a battery). When we want the chemistry to go in the non-spontaneous direction, the cell is an electrolytic cell and requires an external power source to drive the chemistry in a non-spontaneous direction. It is important to note that this is one of the amazing things about electrochemistry. We can use an external voltage to change the direction of a chemical reaction by adjusting the free energy (we essentially get to chose which is lower in free energy the products or reactants). So with an electrochemical cell, we can get a chemical reaction to move in both the spontaneous and the non-spontaneous direction. You are all familiar with this concept but you might not have thought about it as chemistry. When you are using a battery you are running a voltaic electrochemical cell in the spontaneous direction. When it gets to equilibrium the voltage on the battery is zero and no more current flows. However, you can "recharge" the battery by attaching it to an external power supply and forcing the chemistry to go in the reverse direction until you have regenerated the original chemical contents.