Wow. There's a lot of misinformation in the responses that you have already.
Here's the scoop:
The process of wine-making is a pretty interesting process.
The first step, of course, is obtaining the juice from the fruit. In the case of grapes, this means crushing the grapes into "must". Must is basically crushed fruit, stems, seeds, and skins.
Then the juice is strained from the pomace (the solids - stems, skins, etc.) and the pomace pressed to release the rest of the juice.
Next the juice is sterilized, either chemically or by heating. This prevents any unwanted fungus (molds or wild yeasts) from affecting the juice. After this step, the juice is inoculated with the proper yeast to ferment it into wine.
The next step is fermentation, where the yeast digests the fructose and glucose (sugars) in the juice and excrete carbon dioxide and alcohol as waste products. The yeasts keep reproducing and digesting until their environment (the juice/wine) becomes too full of their own wastes (alcohol) and it kills them (often when the alcohol content reaches around 7%, but up to 14% for some yeast strains). Yeah, yeasts are kinda like people in some ways.
After this fermentation step, the wine is "racked" meaning that it's poured off of whatever solids remain and put into casks to age. Eventually the wine is bottled and sold, or is distilled (a process by which the alcohol is boiled off of the wine and then condensed) into brandy.
There are a few steps that I've skipped, and some steps occur more than once, but you should have a general idea of how it's done.
In the US, it's legal to brew your own wine or beer for personal consumption, but illegal to distill it into spirits (such as brandy, whiskey, vodka, etc.) or sell it.