There are seven basic hydroponic system designs, I will give you an overview of these here and then provide you with details in separate articles.
Static Solution or Container Growing
This is a very easy system to build and can generally be made from items you have around your home.
A tank partially filled with nutrients. In this you floating platform made of polystyrene of a similar buoyant material. Cut hole for plant pots into the platform and put the plants into the pot using sand or gravel instead of soil. It is a good idea to aerate the solution with a small air pump. The tank should be painted or have a light blocking material place over it to stop algae growing in the nutrient solution.
Nutrient Thin Film (NFT)
This is a slightly more advanced system and will require at minimum a CF meter to maintain the correct nutrient solution concentration.
A channel is placed on a angle, nutrient is pumped into the top of the channel and flows down the channel into a tank at the bottom to be recirculated again. The plants are grown along the channel. This system is commonly used for commercial growers and can easily convert down to small homemade hydroponic systems.
Aeroponics is a much more advanced system. The plants are suspended above a tank and the nutrients are sprayed in a fine mist over the roots. The spay can be applied continuously or periodically. This method of hydroponics has the best plant aeration and low water consumption.
This is also a very easy system to build. Plants are potted in an inert porous media like a mixture of Perlite and Vermiculite and the pots are placed into a tray which contains a capillary mat saturated in nutrient solution.
Flood and Drain
This also is a very easy system to build and like Container Growing can be made from items found around your home.
The construction of this system consists of a tank of nutrient with a tray filled with a medium like clay granules suspended above it (can easily for the lid of the tank), holes are made into the tray to allow the nutrient solution to flow back into the tank. The plants are planted into which the medium. Nutrients are pumped (or hand poured) into the tray to fill it and then are left to drain back into the tank. There is enough moisture and nutrient left in the tray so that the flooding only needs to be completed periodically.
Run to Waste or Fertigation
Run to Waste is basically applying water and nutrient to plants growing in the ground on a periodic basis, either automated or manually. The quantity applied is very dependant on the moisture loss through the ground and the requirements of the plants. Strictly speaking the plants should be growing in an inert media, only this method can also be used for plants growing in soil
Deep Water Cultivation
This is very similar to the Static Solution System only the plants have their roots suspended directly into the nutrient solution rather than being planted in pots of inert material. The nutrient solution must be properly aerated to avoid root rot.