A hydroponic system is a method of growing plants utilizing little or no soil, allowing you to grow more in a smaller space. Here, I'll be outlining a few different types of hydroponic systems you can build yourself, with their pros and cons. All images are from nosoilsolutions.com.
Wick System
This is one of the most simple system designs available. It is passive, meaning that it will function without power inputs. The plants are placed above the water/nutrient solution, and a wick pulls the liquid up into the growing medium (perlite, vermiculite, etc.) This is great for someone who wants to keep things easy and relatively low maintenance. However, this system is difficult to grow larger plants, as it can be difficult to bring up enough liquid and nutrients to the growing medium. Therefore, its recommended use is to grow smaller plants like herbs.
Deep Water Culture
Deep water culture is a very effective way of growing sizable plants with larger root systems, and to yield plenty of fruit. The plants are held in place in a tray using foam, growing medium, or grow nets, depending on your preference. Plants' roots are then directly placed into nutrient solution, giving them easyaccess to what they need. However, you must properly oxygenate the water using an air pump and air stone. While this design can lead to huge crops, it can be susceptible to disease, so be sure to use filtered water, aerate properly, and keep a close eye out for signs of pathogens.
Drip System
The drip hydroponic system is very efficient and flexible, allowing you to try new things and to conform to your physical environment and plant needs. The nutrient solution is pumped through drip tubes (available for purchase or can be constructed) that are placed into the growing medium of the plants. The flow/drip can be adjusted so that plants are not too dry or too wet. These systems can be circulating - meaning that the nutrient solution returns to the reservoir - or non-circulating, meaning that all solution is absorbed by plants. This design may take a little calibration, but allows you to grow almost anything, which can be very useful for someone wanting to try new plants and foods.