There are six basic types of systems: wick, deep-water culture (DWC), ebb-and-flow, drip method, nutrient-film technique (NFT), and aeroponics. Usually some of indoor vertical farmers used nutrient-film technique (NFT) and ebb and flow. NFT is probably the most popular system for commercial growers. It's what most greenhouses use because of the low water requirements, and it's the most conservative you can be on your water and nutrients.
What is hydroponics? Basically, growing plants without the use of a traditional dirt medium and using a nutrient rich water solution. Those mediums range from fiberglass to sand and from fired clay balls to nothing at all. Several branches of hyrdoponics include aeroponics (using air as the grow medium), aquaponics etc. according to instructables.
For home use I recommend an ebb-and-flow system like this one. There's little maintenance and fewer opportunities to kill your crops. The system is also relatively inexpensive and easy to build. You'll be eating homegrown arugula in as little as two to four weeks. according to popularmechanics.
Hydroponically grown foods not only taste better and are more nutritional, you can change the properties of your food, monitor what goes into your food and pollutes less. You can also grow more in less space. For more information about the steps and materials to be used, just log in popularmechanics.