First off, we should define what Hydroponics is:
Hydroponics is simply put, growing plants in some sort of "medium" with added nutrients. A "medium" is just that. It can be clay, coconut coir, or Perilite. There are more mediums than this basic start, but these is the foundation to understanding Hydroponics.
The most well known "medium" is dirt.
Essentially what we are doing is removing dirt from the horticulture experience, then creating a liquid solution that has all of the nutrients that the plants need. Finally, you will be feeding those plants that solution via their roots.
What are the advantages of Hydroponics? Well that can be as simple, or complex as you want it to be. The basic advantage is you are saving on water and nutrients. Dirt absorbs water like a giant sponge. This in turn then only allows the plant to get a portion of that water.
Here is an example: You are watering a flower in a vase with holes in the bottom. You add water. The next thing you know you have water coming out of the bottom of the vase. Well, that water may be lost(depending on your vase). It may spill all over your sidewalk. Well with hydroponics, instead of pouring water on your plants ever so often you re-use the existing water over and over again. You recirculate the water back through the plants which are grabbing nutrients out of it. If properly done, you really should not have to add water. You just make sure that the pH of the water is in touch, and you have the correct amount of nutrients in the water.
This is an introduction to the basics of Hydroponics. This term is more of an umbrella statement for many different styles of growing plants.
I will not overwhelm you in this post with all of the more specific forms of Hydroponics, that is for another post. :-)
Update: Here is the next post for more information on Hydroponics in a general sense versus dirt.