There are many guides on the Internet telling people how to make high-quality wine with lots of expensive equipment, involving lots of patience. This is not one of them. This is a simple guide. Anyone can make decent wine. I will also discuss advanced topics, but you needn't worry about them if you don't want to.
To make wine, you'll need some juice. I recommend Welch's (concentrate works fine, just reconstitute it properly). Avoid juice with preservatives. You'll need two fermentation vessels of roughly equal size and roughly equal to the amount of juice you have. These can be anything food-grade. Plastic works, glass is better. You will need something to transfer the must from one to the other. You can pour, but a racking hose is much better, and costs less than a dollar (get one at your local hardware store, I recommend starting out with about five feet). I used a turkey baster in a pinch once (it was only a quart of wine, though). You'll also need an airlock that fits on your fermenting vessel. This is to allow the carbon dioxide out while keeping oxygen (causes oxidation and is necessary for vinegar bacteria) and harmful yeast/bacteria out. Saran wrap and a rubber band work fine.
You'll need yeast. Bread yeast will work, but a good wine yeast is so much better. A packet of wine yeast usually costs about a dollar. Find some online. Ask for some yeast from a homebrewer. If you are doing this in secret, say that the yeast is for root beer. Once you've got your yeast, you can use one batch of wine to start another (provided the batch wasn't contaminated). One cup of fermenting wine (I'm not sure if there's enough yeast in finished wine) in one gallon should work.
To start, pour the juice into the fermenting vessel, leaving some head room (it will foam up). Mix some yeast into a small amount of warm water or juice, being sure that it dissolves COMPLETELY! This is critical. Then mix this yeast-juice solution into the must (the must is the wine before it is fermented). This is called pitching the yeast. Attach the airlock. Put it in a dark, cool (60-70 degrees Fahrenheit) place. This is called primary fermentation, because it is where most of the action happens. After about a week, rack it. This means to siphon it into the other (cleaned) container, while leaving behind the dead yeast (lees). Top it up with more of the juice. Avoid splashing, as splashing may cause the wine to oxidize. This is the secondary fermentation.
Rack it whenever a significant layer of sediment occurs. You may bottle it when it looks clear and has stopped releasing carbon dioxide, or when it doesn't taste yeasty (some things won't clear quickly. I lost some hard cider to the dreaded acetobacter while waiting for it to clear. I probably should have just bottled). Rack it into clean bottles or some other containers. Make sure that they are topped up.
The wine may be drank immediately, but will improve significantly with age. I just opened up a bottle of Welch's wine that was a little over a month old. It was excellent! Most sources recommend aging at least six months, so you may want to save a little, to see what it has the potental to be.
With Welch's grape juice, I can bottle my wine in two weeks (I prefer to wait three). Welch's from concentrate yields a tasty dry wine of about 9.5% alcohol by volume. This is not hard to do at all, takes very little time, is tasty, and is cheap (besides, it's self-sufficent, what could be cooler?)
Using real fruit is an option, and will make good wine, but is harder and costs much more (usually). I've read that cyanide is present in small amounts in the pits of apricots, peaches, cherries, plums, apples, and related fruits. So you may want to think twice abount leaving the pits in the must (note however that cyanide is acutely toxic, it doesn't build up in your system). To start, mash the fruit until most of the juice has been extracted. If you want a darker, fuller bodied wine, leave the skins in with the wine for primary fermentation. Otherwise, wring them out, or use a wine press.
If you want to get fancy and measure the amount of sugar in your must, or the potental alcohol yield of your must, you can buy a device called a hydrometer. This is a long (somewhere around eight inches) glass float that measures the weight of the liquid relative to water. From this, the sugar content (by weight) is figured, and the potental alcohol calculated. A printed scale inside lists the specific gravity (SG), the sugar content, and the potental alcohol. You must measure the must before fermentation for an alcohol reading. To get a really accurate reading take the first reading, subtract the final reading, and multiply by 105. That's OG - FG * 105.
Some people like to dilute their juice before fermentation, to make a lighter flavored wine. If you do this, you'll want to add sugar (using a hydrometer preferably) to get a normal alcohol yeild. You may even want to add sugar without diluting. A "normal" store-bought wine is usually 12-14% abv. Yeast rarely ferments to more than 14% abv; it dies of alcohol poisioning. This will result in a sweet wine.
If you want a sweet wine without fussing about getting the sugar just right to kill the yeast and leave enough sugar, you can ferment a dry wine, add a stabilizer, and add the desired amount of sugar.
Some people like to adjust the acid content of their wines. There are acid powders you can buy at your local homebrew supply, as well as pH testing kits. When using low-tannin juice, sometimes tannin is a good thing to add. There are also various yeast energisers and yeast nutrients to speed up fermentation. With grape juice, these are unnecessary.
To make sparkling wine, either add sugar or juice once the wine is clear, and bottle in pressure safe bottles. Be careful, bottle grenades are bad! This is usually allowed to stand until the wine has cleared again. In real Champagne, the bottles are tipped upside down, the yeast collects in the necks, the necks are frozen, the lees are extracted, and are replaced by similar wine or water.
A word on sanitation. Rinse things out. Dust and dirt are bad. Large mold colonies are bad. Try to keep things somewhat clean. Some people use stuff like sulfates and cleaners, but I believe that the must is going to be exposed to contaminents from the air anyway, and you just have to hope that the yeast beats them out (with the alcohol). The only time I would recommend sulfating a wine would be when using fresh fruit, as there may be more wild yeasts on it than in pasturized juice.
Above all, have fun, and make wine that you like. Don't make wine a certain way because that's what you're supposed to do. Try new things. Share this information with a friend. If you have been inspired by this node to make wine, /msg me.