There are many ways to make beer. This is my process.

There are three broad phases to making beer - mashing, boiling, and fermenting.

Each phase is, of course, composed of many smaller tasks. Planning is important.

Mashing

Mashing is the way sugars are extracted from the starch contained in barley (and sometimes wheat or oats, too). Beer is made from barley that has been malted. Raw barley wants to sprout and form a new barley plant. In the malting process, raw barley is allowed to only just begin sprouting, and then it is dried and lightly roasted. This prepares the starches in the kernel, and causes enzymes to become more available, or something. All I know is, if you try to make beer from un-malted barley, you don't get nearly as much sugar out of it. The malting is done beforehand; very few homebrewers have the resources to malt their own barley. You buy your grains already malted - it looks a lot like raw wheat.

To mash, you grind the barley, coarsely. If you grind it to flour, the mash will compact like cement, and it's difficult to get the sugars out at the end, so you want a crush that really just cracks each kernel into 3-4 pieces. There will be some flour, and the barley husk get separated from the kernel during cracking too. The husks also help to keep the mash from compacting.

Once the grain is crushed, you add hot water and mix well - a little water, a little grain, repeat until the grain is all wet. As little water as you can get away with is best - you can always add a little more, but you can't take it away. You want a final temperature between 150 and 160 degrees, so you start with hotter water - how hot depends on the heat losses in your system, and how cold it is where you're brewing. For me, roughly 185 degree water gives me a good mashing temperature. The mash should look like thick porridge when you're done mixing - not soup. I do what's called infusion mashing - hot water + grain, and leave it in an insulated container (the mash tun) for 90 minutes, while the enzymes convert all the starches to sugars. Some people use mash tuns which are actively heated, to maintain better temperature control. Different strokes.

After the 90 minute mash, the lautering/sparging process begins. You want to drain the sweet liquid (called wort) from the grain-bed, while at the same time adding hot water to the top of the grain-bed, without disturbing it too much. The first couple of liters of wort collected are quite cloudy, so add them back into the the mash-tun, and allow the natural filter bed of the barley husks to clarify the wort somewhat. Thus you gradually rinse or leech the sugars, from the grain-bed into the sweet liquid wort. You can't rush this phase (well, you can, but you'll leave sugar behind if you do). Drain slowly by restricting the outflow, and be patient - it should take 45 minute or so at least. I almost always drink a half-cup or so of the first sweet wort - best taste in the world!

I use a "total-amount-of-water" method of sparging - I use a fixed amount of water, and then stop. Some people sparge until the amount of sugar content in the outflow falls below some threshold. That's more work, for bugger-all difference, so I don't bother (in general, brewing is quite a forgiving process). So my process is, heat water for the mash-in, then when the mash time is nearly up, heat more water for the sparge. Wort is collected in plastic buckets, and when the last of the sparge water is out of the kettle, the wort goes back in the kettle, to be brought up to a boil. One heat source is all I need.

I use a method of hop addition called "first wort hopping" - it's not very common, but it works well for me. I put a quantity of hops into the first wort-collection bucket, and let the wort (hot from the mash-tun) steep in these hops while the lautering continues. This hour or more of steeping, at lower-than-boiling temperatures (say, 140's) extracts hop flavours into the wort, which survive the coming boil somehow. Other brewers add hops late in the boil for the same effect. Different strokes again.

Spent grain goes in the compost pile, or to the chickens. Collected wort goes in the kettle for the boil.

Boiling

The wort is boiled for several reasons. Foremost is sanitation - barley sugars are delicious to many microbes, but we only want yeast eating our sugars. The next reason is to extract bitterness from the hops (beer without any bitterness is undrinkably bland). And finally, a good long boil is necessary for the clarity of the beer - coagulations of proteins or something, I don't know - but your beer will clear better and faster if it has a long hard boil.

For reasons I don't understand, first-wort hops seem to contribute almost nothing to bitterness (although lots of flavour). So, once the wort is boiling, extra bittering hops are added and boiled for at least 60 minutes. When the wort first comes to a boil, it will want to foam up and out of the vessel, so you must watch it very closely, and back off the heat when the foam rises (otherwise you'll have a sticky carmelised mess all over your kettle). Once a full boil is acheived, the foam will stop rising and the 60-minute bittering hops can be added.

After the boil, the wort must be cooled to roughly room temperature, so that the yeast we're going to add don't all die. It takes forever for wort to cool on its own, and we want the yeasts to start working as soon as possible, so it is necessary to forcibly cool the wort. I use a simple immersion chiller - a coiled length of copper tubing through which cool water is forced. The first water out of the chiller is quite hot, so I capture it and use it for cleaning the mash tun and other equipment.

So, the kettle is covered and the wort cooled. In my system, this takes about 30 minutes. The kettle is covered from this point on, to keep out dust, leaves, bugs, whatever. Sanitation is critical after the heat of the boil is gone, and before the yeast have dominated (the ethanol they produce discourages late-coming microbes from gaining any sizable populations).

Fermenting

The cooled wort is transferred into the fermentation vessel - I use two glass carboys (big jugs). I have a strainer in the kettle that keeps the spent hops behind. Add the yeast, pop on an airlock, and put it in a cool dark place for about 2 weeks. Vigourous fermentation can cause foam to overflow the container, so leave plenty of headroom and keep temperatures cool to prevent this. In warm weather I sometimes put an old t-shirt over the carboy, and put the whole thing in a tub with an inch of water in the bottom. The water wicks up the t-shirt and keeps the carboy a few degrees cooler through evaporation.

When fermentation finishes, you have beer! But it is flat. So you add a measured amount of sugar, and bottle. That extra sugar re-starts a small fermentation cycle in the bottle, which gives the carbonation. That's why it's a measured amount of sugar - too much, and the bottles will explode. So make sure the fermentation is done first, then add just enough sugar.

That's all, really. It's ready to drink quite quickly, but for best flavour, give it at least a month in the bottle. The stronger the beer, the longer you should let it mellow. Beer will keep for at least a couple of years, if stored in a cool dark place (unless using plastic bottles - these are okay, but the beer will gradually oxidise, and will not last years). Stronger beers age better, and sometimes need the extra time for best flavour.

Postscript

I wrote the above while waiting for the mash. This brew day I had a slight complication - a stuck sparge. Usually I only have to worry about a stuck sparge when using a lot of wheat in the grist - it can be gluey. This batch I used 7 kg Munich barley and 1 kg malted wheat, so I didn't think I'd need to worry about a stuck sparge, but it happened. When gathering the first few liters of wort and re-circulating, I opened the flow full-bore for a few seconds, before clamping it down. Usually this is okay, but if you do it with a high wheat content grist, you can jam the sparge, so you have to be careful to drain slowly at all times. I was careless because I wasn't using much wheat. By draining fast, it caused a portion of the mash grain-bed to compact tightly, restricting flow completely. So, after draining only about ten liters, I had to add a little hot water and re-stir the entire mash, breaking up the layer at the bottom. I let this settle for 20 minutes, and re-started the sparge. No harm done, except for the loss of a lot of time.

Feels good. Feels really good.