In layman's terms, hydroponics is the science of growing plants without soil-- although the plants may or may not be suspended in a solid medium such as gravel, or expanded clay balls.
Soil retains minerals and nutrients, which "feed" flora, as we all know. Plant roots can't absorb dirt, however; when water passes through soil, it dissolves and collects some of the nutrient particles embedded. This "food" solution is absorbable as a liquid. As you can see, the soil itself is not an integral part of a plant's feeding cycle-- it is simply a stabilizer for the roots, and a convenient filter.
Why eliminate the soil?
Plants breathe air, just like humans. School children are taught a simple lesson: plants take in carbon dioxide, and release oxygen. The entire plant-- not just leafy material-- contributes to this process.
If not properly maintained, soil can retain too much moisture, effectively suffocating ("drowning") a plant's root system. Alternatively, if the soil doesn't contain enough moisture, the plant will be unable to absorb the nutrients it needs to survive.
The roots of a hydroponic plant have constant access to both air and water, and it can be much easier to maintain that balance since the roots are typically visible.
The average plant needs at least five things to survive. Air, water, nutrients, minerals, and light. So long as you can provide these things in plenty, your plants should stay healthy.
Growing your own food can be a rewarding experience. It's a good way to save money on pesticide-free produce, and you'll know it wasn't shipped from a third-world serf farm supporting bad business. If your hydroponic system is indoors, you can grow food during the off-season in winter, too.
That being said, there may be more efficient systems out there for the home grower. I created this instructable to inform, more than anything.
After all, if anything's worth doing, it's worth doing right. Gotta do your research, kids.