For the basic home gym, all you need is a set of dumb-bells, an exercise ball, a chin-up bar and a pair of running shoes. With those, you can cover all the bases.
For a bigger home gym, get an adjustable cable pulley machine as well. Highly versatile, a cable pulley machine will allow you to work most of the core muscle groups (avoid multi-gyms as they won't work all your muscles as effectively).
For aerobic activity, a rowing machine, a running machine and a bike are all very useful, as can be a cross-trainer.
When it comes to lifting weights, the less machine work the better. Lifting free weights will encourage the development of core strength as more muscles come into play, so a good collection of dumb-bells is essential for the home gym.