What makes trains move?

A propulsion car, usually a locomotive today.

The first trains used people power, horse power, and gravity. Sail power was even tried in the early 1800's in the US. Then steam locomotives, followed by diesel locomotives, followed by electric locomotives, - Today, electric is the most common for passenger trains with diesel the most common for freight trains. Today, some trains distribute their propulsion along multiple cars and don't need a locomotive (referred to as MU, for multiple unit).