A hyperdrive does the following: if a hyperdrive-equipped ship is outside the 'hyper limit' of all stars (some distance depending on the size of the star, you can check it somewhere) it can perform the order 'activate hyperdrive'. This increases its maximum speed to 10 fold (if I remember correctly) its normal maximum speed. There are no limits to this, it uses no extra fuel and has no combat penalties (as far as I know). The only restriction is that if your ship ever enters the hyperlimit of a star while the hyperdrive is engaged, the game will produce an interrupt every 5 seconds and your ship will be unable to move until you deactivate its hyperdrive.
There are two main uses for hyperdrives, as far as I have been able to tell. If you have some way of coaxing an enemy to leave the hyper limit, your combat ships would suddenly have a massive speed advantage. Secondly, some systems, especially those with multiple suns (binary, etc), have vastly spaced planets or planetary systems either outside the hyperlimit or, in the case of multiple suns, with a gap between the two stars' hyperlimits. If these systems do not have convenient Lagrange Points, then hyperdrive equipped freighters can significantly shorten the travel time to these planets or planetary systems.
Hyperdrives are really annoying, though, since you have to switch them on and off manually or at a waypoint every time. In truth, I don't think many players use them, simply because they are so irritating.