Hey wordwiz, there are many kinds of Hydroponics.
1. DWC (Deep Water Culture)
This is a method that was used by the Aztecs in their famous floating water gardens. The modern setup is simple. It's a Rubbermaid container with holes in the lid that you put net pots into with gravel or hydroton that your seedling fits into. The roots hang down into the water in the tub that is agitated by large "bubblers" that put oxygen into the water. You can do a search on Hydroponic DWC and look at a bunch of them. If you decide on this one, let me know and I can walk you through a build.
2. Drip Hydro
This one is also simple. Basically, you have a tube or trough that you put your plants into and angle so that the drip you give them drains back into the reservoir. It also uses net pots like the DWC. It looks like willard has this one down pat and could prolly help you build one. If not, I'd be glad to.
3. NFT (Nutrient Flow Technique)
This one was originally developed by the US Army to feed it's troops in the Aleutian Islands. Nothing grew there, so all food had to be shipped at an outrageous cost of money and lives. It's similar to the drip system and in fact some people use it with a drip to make a Hybrid system combining the two. The water is pumped to the top of the troughs or tubes and drains back to the reservoir, past the plant roots. I can help you set this one up as well. It's not a good inside method. It has a bad habit of leaking water all over the place. I've seen a few of them ruin a lot of carpet.
4. Ebb and Flow (Flood and Drain)
This is what I use in my hydro room inside. It's a system of tubs full of gravel or hydroton with the plants buried in it. The nutrient water pumps to fill the containers and cycles back into the reservoir. It's considered the most efficient of the hydro methods, but is also the most costly and difficult to set up. If you look at my Blog, you'll see one that is almost 20 years old. They work forever with only a pump replacement once in awhile. Let me know if you want to set up one of these.
5. Last, but not least, there is Aeroponics
This method isn't considered a very cost effective method because of a bad habit of clogging the spray heads, but it is recognized as the absolute best of the best in hydro growing. The roots are suspended in air and sprayed with a very fine mist of nutrient water. It maximizes the oxygen/nutrient absorption and plants love it.
These are very basic descriptions of the various types of Hydroponic systems, and 99% if what you need to know isn't described here. I'm intimately familiar with each of the systems and can help you build whatever system you decide on.
If you want to really learn about Hydroponic Food Production, then I would strongly suggest that you buy the book:
Hydroponic Food Production by Dr. Howard Resh. This book is called "The bible of Hydroponics" by many people and will describe and show you many, many ways to build systems that I've described above. The photos and diagrams in the book are worth a million.
Good luck to you. Let me know if I can be of help.