What is a cathode?

The electrode at which a reduction reaction occurs.


Physics
A cathode is the element of an electron tube from which electrons flow. Back in the day, current was thought to be positive, but we understand things differently now. In the "old school" version, the cathode was the element into which positive charges flowed. But that's usually not what is taught for the simple reason that it's a classical physics approach and isn't at all as useful as the idea that the cathode is the element from which electrons leave to go to the plate.
In a two-element tube, the cathode has a compliment called the anode, or plate. Electrons flow from the cathode to the anode. Not the other way. There are tubes with three, four, and more elements, but they have these two basic elements.
Chemistry
The cathode is the electrode of a polarized electrical device, such as a galvanic cell, out of which positive electric current flows. In a battery like the one in a vehicle, the positive terminal is the cathode. The car battery, which is a lead-acid battery, also has an electrode that is the compliment to the cathode: the anode. It's the negative terminal of the battery, and positive electric current flows into this electrode.
The Wikipedia article on the cathode covers both the chemistry application of the term as well as the physics application, and a link is provided.