Legality
Yes, it is legal folks. In 1979, the Federal government waived the requirement for obtaining a permit for home winemaking. A household of two or more adults is allowed to make a maximum of 200 gallons of wine a year, or 100 gallons a year for a household with a single adult. You are not allowed to sell it however. If you are looking to sell wine you make, you'll need to apply with the TTB to be a bonded winery, and pay taxes on your wine production. You may want to check to see if your state has any other restrictions on home winemaking.
How is wine made
Wine is one of the simplest alcoholic beverages to make. You start with grapes, or some sort of juice with enough sugar in it, and toss in some yeast. The yeast consumes the sugar in the grapes/juice and produces CO2 and alcohol. The CO2 gas just bubbles up from the grapes/juice (which is called a "must" at this point) and go into the air. After the sugar is consumed, the yeast start to die off, and the dead yeast and other particles in the must start to precipitate and fall to the bottom of the container it is in. What you have then is actual wine, and the yeast did all the work for you! There are other things you can do with it after that point, depending on the style of wine you are trying to make, but essentially that is it.
How to get started
First you will need some equipment. The equipment will consist of a container (bucket) to do your fermentation in, another container to rack (siphon) the wine in your bucket into after most of the fermentation is done with (this second container will help protect the wine from oxygen, and frees up your bucket for other fermentations). You'll also need a siphon, tubing, some chemicals (preservatives, cleaners and concoctions to help particles fall out of the wine), and some bottles. You can buy an equipment kit that will contain some or all of these from local or online home winemaking supply stores.
Secondly, you'll need something to make wine out of. You can use a recipe for some sort of fruit wine, there are a lot of them out of there, including some recipes on this site. Or you can make wine out of a kit. Kit wines contain concentrated grape juice (which you have to water down to get the volume right), yeast, and all the chemicals and other additives you'll need to finish the wine off. Or, you can make wine from grapes (like most of the wine sold at the grocery stores are).
That's about it! Easy, huh? Well, I do have to warn you, making wine at home can become more than a hobby, it can become a passion. You may want to expand your knowledge and experience to make the best wines you can. You might even get to the point where you want to send your wines into competitions to see how good you are compared to other home winemakers. And you may even get so into this "hobby" that you actually consider the idea of starting your own commerical winery, to compete with the other big names in the industry. Think I'm exaggerating? Well, I have started having these thoughts myself, and I've spoken with a lot of commercial winemakers and winery owners, and a large percentage of them started this way. Remember, you have been warned. :-)