C.E. Evans wrote:
Warp drive is just another name for a FTL drive. The full intricacies of how it works have remained vague onscreen, although later Trek shows have often mentioned "subspace fields" or "warp fields," but otherwise it's just a device to get our heroes where they need to go.
Star Trek (at least TNG and later) very clearly defined the theory behind the technology. We've seen a ton of on screen footage and images demonstrating it, but the nitty gritty is also outlined in pretty much every official tech manual.
Someone actually put together a very good Youtube video explaining a while back: