ok, here goes.
The first rule of brewing: Sterilization is very important. I use a bleach solution in the sink, although you can buy other types of sanitation chemicals if you want. First clean out the sink with soap and water and make sure there aren't any bits of food or anything in it. It doesn't take much bleach, and I probably use more than I need with around 1 cup of bleach per 2 gallons of water. Mix that around and put any utensils that are going to be touching the beer in the water to soak for at least 5 or 10 minutes. When you drain the water, it also leaves a clean surface to store your utensils. Be sure to rinse the utensils in clean water before putting them in the brew.
I brew 5 gallons of beer per batch, and for that you'll need at least a 4 gallon kettle to boil water in. Stainless steel is highly recommended because it won't rust or corrode and put bad flavors in your brew. A brew store will sell you one of these for about 40 bucks, but I saw one at United supermarket the other day for 15. I dunno if it was as high quality as this one is, though.
This is a 6.5 gallon glass carboy. A carboy is a fancy name for big glass jug. It is for fermenting the beer. Glass is better than a plastic jug, which you can also use, for a couple of reasons. First, it won't scratch easily, and bacteria can live in scratches even after cleaning. Second, if you make a dark beer in a plastic jug, and then want to make a light beer, you gotta scrub the hell out of your jug to get all the dark beer flavor off of it. Or buy a new one for light brews.
This is a pail in which to store ale. Notice the tap at the bottom. That is for bottling. After fermentation you will siphon beer from the glass fermentation tank into this plastic jug, then pour into bottles for another round of aging.
This is a bottling doodad... you can buy caps at a brew store and also bottles for it, or you can go pick up some Dos Equis at the liquor store and reuse those. The thing is, you gotta get pop-off bottles, not twist off, or this won't work. Alternatively, you can use plastic coke bottles, old 40 oz liquor bottles, or anything that is resealable to an airtight state.
This is priming sugar, also known as corn sugar or dextrose. It gives the yeast something to eat, which promotes body and alcohol content. The reason you don't use normal table sugar is that the yeasts will eat that and give the beer a very cidery taste. Normally you use this at bottling, but sometimes I use a little bit before fermentation too.
This blurry picture is of bittering hops. They go into the brew while it's boiling to give it a bitterness. There are also aromatic hops, which go in at the very end of the boil. I'll talk about those later.
A floating thermometer is handy for keeping the beer at 150 degrees while you are steeping your grains. You don't want it higher than that until later when you boil. The floating thermometer isn't strictly necessary, but handy.
This is malt extract. It's almost pure sugar, and a lot like molasses. It is food for the yeast to eat during fermentation.
Store your beer in a closet and make sure you have a way for extra carbon dioxide to escape. The thing sticking out of the plug is an airlock. It lets CO2 out without letting any oxygen in. A purple glittering hippie pillow to oversee your brew is optional, but recommended.
OK, here's the basic steps to brew beer. Your recipe may vary, but a lot of them are done this way. This is for a 5 gallon recipe.
You will go to the brew store and buy some grains, malt extract, yeast, hops, and anything else you need. Which grains you use depends on the kind of beer you are brewing. Grains give the water color and flavor, and also acts as food for the yeast to eat. Typical kinds are barley, wheat, oats, etc.
Put the grains into a cheesecloth. Bring 3 gallons of bottled water (or tap water if you want, but it might give your beer weird flavors) to 150 degrees. Steep the grains in the cheesecloth like it was tea for about 15 minutes. The water will take on a color depending on the grains you used.
After 15 minutes at 150 degrees, remove the grains and bring the water to a boil. When it reaches a rolling boil, remove it from the burner and pour in your malt extract, stirring constantly. It's very beneficial to have someone to help you at this point. After putting in the extract, put the pot back on the burner and bring back to a rolling boil. Add in your bittering hops and let it boil vigorously for 1 hour. During the last 5 minutes or so, add the aromatic (flavoring) hops if the recipe calls for any. Stir it a little bit, but the hops will dissolve in the hot water pretty easily.
What you've just made is called wort (pronounced 'wert') and amounts to immature beer.
After an hour of boiling, remove from heat and let it stand, covered, until the beer reaches about 120 degrees F. Add 2 gallons of cool water to your fermentation tank and pour the wort on top of it. Don't worry if some solids go into the tank; they'll settle out of solution. Shake the tank vigorously to aerate it, the yeast will appreciate it. Leave the tank until it is cool to the touch, then add your yeast. You can use liquid yeast, which cost about 6 bucks for one batch, or dry brewers yeast (don't use bread yeast like you get in the grocery store!) which costs about 99 cents for 1 batch. I've used the dry yeast with good results, but they tell me that the liquid yeast is very good. The act of putting yeast into the beer is called "pitching" the yeast.
Place the fermenter in a cool, dark place out of direct sunlight (I use the back bedroom closet) for 2 weeks, or as dictated by the recipe. Don't shake it or disturb it. It is normal for some gas to escape the tank while the yeast go to work. Sometimes even some liquid spews out, but don't worry about that other than to cover anything you don't want wet inside the closet.
After 2 weeks, use a siphoning tube and caning rack (a hard plastic tube so the siphoning tube won't curl up in the bottom of your fermenter and you can get all the beer) to drain the wort into your bottling jug. Mix up 3/4 cup of the priming sugar in 1 cup of boiling water and stir until dissolved, then add that to your bottling bucket.
Make sure you've sanitized some bottles with a bleach solution or something, and rinsed them thoroughly. They don't have to be 100% dry when you bottle, but make sure no bleach odor is present in them. Pour wort into the bottles to about 1 inch below their lip, then seal them tightly. Put the bottles into the closet for at least another 2 weeks, then chill and enjoy!
If you taste the wort at bottling, it should taste like flat, immature beer. Most of the carbonation occurs in the bottle part of the aging process. The solids that settle out of the fermentation tank are not bad for you, in fact they are full of B vitamins. So don't worry if some get into the bottles, though you might not want them in your glass.
Although the beer is ready to drink in 2 weeks, extra aging improves the quality of most beers. I had some that was in its bottle for 8 weeks and it was excellent, as good as I would expect from the Guiness tap at a bar.