A Hyperdrive is a form of Star Drive that propels a spacecraft into subspace or hyperspace. In most cases, the hyperdrive provides the spaceship with propulsion while in hyperspace, though it's possible the hyperdrive itself might just shift you into the other reality where your normal drives propel the vessel much as they did in normal space - such as in cases where the hyperdrive just makes a wormhole connecting normal space to hyper space.
The difference between a hyperdrive and a jump drive is that the ship equipped with a hyperdrive experiences some amount of time in hyperspace. The "jump" of a jump drive has no duration, or a duration measured in microseconds, whereas the hyperdrive takes minutes, hours, or days to get where it's going. Travel in hyperspace is presented as being some form of FTL, but the exact nature of the subspace or hyperspace involved may have some impact on how it functions.
Hyperdrive travel is often depicted as "blind" or partially so. Ships in hyperspace cannot interact with nor detect ships in realspace, and vise-versa. In this way, Hyperdrive may allow for Stealth In Space, as ships suddenly appear nearby, having dropped out of hyperspace without warning. In some settings, a ship in hyperspace can pass through or ignore matter in real space - making the hyperdrive effectively just a slower form of jump drive. In other settings, matter or gravity has some sort of analog or affect in hyperspace, and ships using hyperdrive have to be careful not to get too close to the gravity well of a large object. Such a ship has to still chart a course through the void of space, it just has the luxury of doing so faster than light.