Hydroponic gardening can be VERY complicated, with computers and sensors controlling everything from watering cycles to nutrient strength and the amount of light that the plants receive.
On the other hand, hydroponics can also be incredibly simple, a hand watered bucket of sand with a single plant is also a method of hydroponic gardening. Most hobby oriented hydroponics systems are somewhere between the two extremes mentioned above.
The “average” home hydroponic system usually consists of a few basic parts: a growing tray, a reservoir, a simple timer controlled submersible pump to water the plants and an air pump and air stone to oxygenate the nutrient solution. Of course, light (either natural or artificial) is also required.