The really short version: You take grains, like barley, and you grind them up into bits and dump them into some very hot water. The enzymes present in the grains will take the starches inside the grains and convert them into sugars. The process of turning the starches of the grains into sugars is called, "mashing." Once you're done mashing, then you move the liquid (which is then called "wort") into a boil kettle and boil it down, thickening it up and adding some hops during the boil to give the beer some bitterness and also to give it some nice flavors and aromas. Then you chill the wort to around room temperature and then add the yeast. Once the yeast is added then the wort is called "beer." Let the beer sit and ferment for a couple of weeks and then separate the beer from the dead yeast and then chill and carbonate for a few days and then serve.
The longer version:
Brewing yeast takes sugar and converts it into alcohol and CO2 and some other tasty flavor components. This changing of sugar to alcohol and CO2 is called "fermentation." So you need to get brewing yeast and sugar together in a water solution and blammo, you will have beer.
Grains contain starches, which can be converted into the sugar that beer yeasts love to eat (which are simple "short-chain" sugars) and also some complex long-chain sugars that the beer yeasts don't eat but still taste sweet to us. Enzymes contained inside the grains can convert those starches for you. First, you grind up the grains so that the starches inside the grains are better exposed to the enzymes. Then, you mix the grains into very warm water, at a temperature most conducive to allowing the enzymes to convert the starch into sugars. This is called, "mashing." The usual temperature range for mashing is between 148° and 156°. If you mash at the lower range, say between 148° and 150°, you'll get more short-chain sugars that are more easily fermented by the yeasts, which means your beer will have more alcohol in it and will have less sugars that won't be fermented, so your beer will taste "thin-bodied" or more watery. If you mash at the higher end of the range, say between 152° and 156°, you'll get more complex long-chain sugars which the yeasts will leave alone and thus you'll get a beer with less alcohol, but more body because of those sugars just hanging out in the beer not being converted into alcohol. The whole mashing process usually takes between 30 minutes to an hour. The pH of the mash is usually very important too. The mash pH should be between 5.1 and 5.6 and the pH affects things too, but that topic is beyond the scope of this, so let's just move on.
So after the mashing process is done, brewers often raise the temperature of the mash above 160° which kills off (or "denatures") the enzymes, and it also makes the sugary liquid (which is called "wort" at this point) a little easier flowing. So the brewer transfers the wort into a boil kettle and then runs some fresh water through the grains again just to be sure to flush out as many sugars as they can get. This rinsing of the grains is called, "sparging" and it can be done in a few different ways. One popular way of sparging is called "fly sparging" and fly sparging is adding fresh water to the top of the grain bed while the wort is being drained from the bottom of the bed at the same rate. This method kinda acts like a piston, pushing the wort from the top down through the bottom. How fast you sparge can greatly affect how much sugars you leave behind, with a slower sparge giving you a better efficiency. Opinions vary, but a good starting point for sparging is about a quart per minute, so this can take quite a while, depending on how much beer you're trying to make.
And then once all of the wort is transferred to the boil kettle, you begin to boil and once it starts boiling then you add hops at different intervals. Add hops early on in the boil and that will add bitterness to the wort, but less flavor and aroma of the hops. Add it later on in the boil and it will add less bitterness but more flavor and aroma. Boiling also reduces the water volume and thus thickens the wort up a good deal, as well as affects the flavor of the sugars in a few ways.
And then once you're done boiling then you need to chill the wort down to a temperature that the yeast will like, usually around room temperature. This chilling can be done while the wort is being transferred to a fermentation tank, via a "heat exchanger." Also, yeasts require that the wort is properly oxygenated and the boiling process removes the oxygen from the wort, so you have to get oxygen back into the wort somehow. You can pump in O2 while the wort is being transferred into the fermentation tank, or you can simply shake the wort up and the sloshing action will oxygenate the beer. Then you add the yeasts. Once the yeast is added then the wort is now deemed "beer." Keep the beer at a stable temperature and out of sunlight for a few weeks and then transfer the beer away from the dead yeast and put that beer in a new tank that you can add CO2 to which will carbonate the beer. Once the carbonation process is complete you can chill and serve.
