What are they?

Ion channels are minute resistors embedded in the cell's membrane, they individually pass electrical currents in the order of 5x10^-12 Amperes. So that's about 1/100,000 of a 1/millionth of the current in a light bulb. Small, but not bad for a single protein.

What do they do?

They enable a cell to communicate with other cells and with the rest of the body. They generate variable electrical signals in response to chemicals and received electrical signals. These signals are then either passed on to other cells, or initiate some sort of cell activity, such as secretion of hormones or contraction.

How are they studied?

With the patch-clamp technique!! Pressing a tiny glass electrode up against the surface of a cell allows you measure the currents flowing through channels, and the potential difference (voltage) across the membranes surface. Typically, a cell has a potential difference between it's inside and the outside of around - 0.1V (conventionally, the cell is said to be negative on the inside). So patch-clamp recorders are very sophisticated ammeters and voltmeters. (And very expensive too!) The inventors the patch clamp technique received the Nobel Prize about 10 years ago, although they invented (developed ? ) it about 20 years ago. Sakmann and Neher circuit. They published an excellent article on this work in the Scientific American in March 1992 (Neher, E., Sakmann, B. The Patch Clamp Technique. Scientific American. March 1992. 44-51).