There's an additional dimension that's neither space nor time. Under normal circumstances, we do not move in either direction along it. A hyperdrive allows for movement along that dimension.
This hyper dimension is broken into bands. Normal space is a 4 dimensional hyperplane through the 5 dimensional hyperspace. The hyperspace bands are 5 dimensional 'slabs' parallel to normal space with 'gaps' in between. The gaps are the hyper walls. The alpha wall separates the alpha band from normal space, the beta wall separates the alpha band from the beta band, and so on. There's some ambiguity whether there are a finite number of discrete 4 dimensional hyperplanes within each level or whether it consists of a continuous 5d hypervolume with infinitely many 4d hyperplanes.
Within a band a hyperdrive can move you up and down the hyper dimension, and it can jump you over the gaps. There's a notion of a velocity along the hyper dimension and this seems to apply to the jumps over the gaps as well.
Impeller drives and reaction drives (and presumably spider drives as well) all work the same in most of hyperspace as they do in normal space. The exception is Gravity Waves which cause impeller drives to blow up, and can even damage ships without any gravitic systems if navigated incorrectly. I don't think there's been any suggestion of how they might interact with spider drives.
Gravity waves are distortions of spacetime that occur in the portions of hyperspace other than normal space. They are generally linear structures with a defined 'width' and portions of that width are flowing on one direction or the other along it. These streams within the wave carry charge particles and by using a hyper sail within a stream moving the right direction, and ship can move along it can achieve much higher acceleration than under impeller outside the wave.
So generally speaking, ships move into hyperspace using their hyperdrive, move to the nearest gravity wave under impeller, switch to hyper sail and find a current int he wave going the direction they want, and move along it. If they need to change to another wave, they have to move out of it, switch to impellers, and then traverse the intervening space.
Wormholes are special gravity waves that intersect normal space rather than being confined to hyperspace. The result is that if you go to the right location in normal space, turn on the hyper sails, point in the right direction, and turn on your hyperdrive, you don't move into hyperspace like normal, but instead jump to a distant point in normal space without passing through the intervening space.