As I understand it, dilithium crystals control the matter/antimatter reaction somehow. Zephram Cochrane's first flight used a ship powered by a nuclear reactor, I think.

The warp field is made with an array of "warp coils" which apparently require the large amount of energy from the matter/antimatter reaction to create a subspace distortion field that allows warp travel. What is subspace? It's never adequately explained, but sometimes monsters come out of it and kidnap you to a pocket dimension where they rearrange your organs.