This is a method of brewing beer with an emphasis on getting it done inexpensively but without sacrificing quality.

[edit] Ingredients

malt
hops
water
yeast

[edit] Steps

Choose the grains. Malted barley is the grain of choice for most beer. Other types of grains can be added (oats, rye, wheat, etc.) but barley it the best to start off with. Malted barley is made by growing the barley then drying/cooking it before the seed splits. The barley is processed in different ways to get different characteristics. Typically you want around 9-10 lbs base malt per 5 gallons. 2-row British pale malt is great to start off with. Add 1-2 lbs of specialty grains to get some good flavor. Lighter crystals are good to add sweetness. Toasted malts will create more of a malty body. Taste the grain as you formulate the recipe. This is a great indicator on the quality of the grain.
Mill the grain. The grain needs to be cracked open to gain access to the goodies inside. The home brew shop should have a mill to use if not one can be purchased or constructed. Usually these are two rollers about 1/8" apart that the grain is fed into.
Make the mash tun. The grain contains mostly starch that needs to be converted to sugar so that the yeast will have something to ferment. The enzymes in the grain are going to do this and they do it well. The mash tun will hold the grain at temperature for 1-2 hours. The mash tun can be made in different ways but a 6 gallon bucket with a false bottom (a mesh at the bottom with drainage) will be the cheapest.
Start mashing. For every 1 pound of grain heat 1 quart (1/4 gallon) of water to 170 degrees. Pour the 170 degree water along with the grains stirring as you go. You want to uniformly add the grain along with the water so it doesn't get too hot. The temperature should be anywhere from 148-158 degrees. If it is somewhere in the middle you're golden. Insulate the mash tun by wrapping it up in a blanket or sleeping bag or jackets. While you are mashing, start heating up 2 quarts (1/2 gallon) of water per pound.
Add the wort. God bless the sweet wort. After about an hour the conversion from starch to sugar will be complete. You can test this with iodine. Put a small amount of wort on a white surface (like a plate) and add a drop of iodine. If it turns black starches still exist and it needs to sit longer. NOTE: Discard the wort with the idodine do not put it back in the wort. If it doesn't change color at all the conversion is complete. Start draining off the wort from the grain bed.
Perform the sparging. Sparging is the process of rinsing the wort off of the grain bed. The best no non-sense method it to do it in 2 steps. When the wort is done draining, add half the sparge water at 180-190 degrees and let it sit 20 minutes. Drain. Then do it again. You want around 6 1/2 gallons of wort in the end.
Add the hops. Hops are a green flower from a hop vine. They add bitterness to the beer which provides a balance against the sweetness of the sugars. Hops also add aroma and flavor. There are many different varieties. Fuggle, Kent Golding, and Cascade are good varieties you can't go wrong with. The higher the alpha acid percentage the more bitter it will be. 4-5% is about average and 10-12% is high. There are calculations to figure out the IBU's (Internation Bittering Units) to get a more exact bittering estimate. 10-20 IBU is low to average bitterness (think light lager) 40 is moderately high (think pale ale) and 50-60 is very bitter (think IPA). The longer the hop boils the more bitterness will be extracted. http://byo.com/feature/467.html
Meanwhile, get the fermentors. The cheapest fermentor is a 6 gallon bucket with an airlock on the top. An airlock is a device that only lets air out and not in. They should be scrubbed clean of any crud from the previous ferment and sanitized with Iodophor solution (check your local homebrew shop) or a bleach solution, two tablespoons per 5 gallons of COLD water. Let this sit for 20 minutes then rinse with clean filtered water twice. This sanitizing step is best done while the wort is boiling, otherwise its an hour spent waiting not getting anything else accomplished.
Boil. Boil the wort for at least 1 hour at a rolling boil. The more vigorous a boil the better. Once the wort is boiling add the flavoring hops. 1 oz of pellets is good to start with. Make sure to stir wort while it is getting up to temperature. While the wort is boiling make sure the fermentors are clean and ready. 10-15 minutes before the end of the hour add the flavoring hops usually about 1/2 oz. 5 minutes from the end add the aroma hops 1/2 oz.
Chill the wort. You can purchase a wort chiller but it is unecessary if you have boiled less than 3 gallons. The chiller will cool the wort down to 70-75 so you can pitch the yeast. Without a chiller you can still get good results by placing the wort boiling pot into an ice bath, either in a large sink, or the bathtub. It is imperative to chill the boiling wort down to below 80 degrees as fast as possible. To speed things up, while it is chilling you can gently stir it without splashing. Aerating hot wort can lead to long term storage instablility problems and is best to avoid.
Fill the fermentor. Pour the finished wort through a strainer to remove the hops into the fermentation bucket. If needed, add only pure clean water to top the fermenter off at a little over 5 gallons. Seal it up and shake the heck out of it to incorporate much-needed oxygen into the wort. This is safe to do now that the wort is chilled and very important since it provides the initial oxygen for anaerobic yeast reproduction. When the wort is 70-75 degrees pitch the yeast.
Choose the yeast. White Labs liquid yeast is recommended. You get a lot of those little yeasties for your money and a starter is not necessary.
Transfer to a clean secondary. After 1-2 weeks of primary fermentation transfer to a clean and sanitized secondary to clear the beer up and let it condition. Use a sanitized siphon to get the beer from the primary to the secondary. The beer will have most of the alcohol in it already so it will be more resistant to nasties in the air. Tips: Avoid sucking on the siphon and getting mouth germs into the beer. Definitely avoid splashing at this stage, since alcohol is easily oxidized and will make the beer funky. If you can pump some CO2 gas (paintball cartridge size will do) into and fill the secondary beforehand you are an ace and will have the best beer transfer possible. Be careful, but realize you aren't getting ready to do surgery or anything.
Keg the beer. It is so much easier than bottling. It is more money but in the end, it saves you a lot of time. Soda kegs are used with a CO2 tank to pressurize. Clean and sanitize the keg. Fill it with CO2 gas to provide a protective blanket (CO2 is heavier than Oxygen and sinks, thus preventing the alcohol present from oxidizing) and gently siphon the beer in. Seal the lid and chill it in your fridge down to at least 40 degrees. (this is not necessary to chill it but the colder it is the more gas will dissolve into it). Hook it up to the tank to pressurize to 20 psi. It will take about 1 hour of rolling the keg around under pressure (while purging off excess pressure that builds up) to get the CO2 in the beer if you want to drink it right away. Otherwise, let it sit for a couple days at 30 psi.
Serve. Lower the pressure to 12 - 15 psi by purging off some gas from the gas inlet. The easiest way to serve is using a cold plate in a cooler. No fridge and it is portable. Run the lines through the cold plate and throw some ice on it. Then run a line with a tap out of the cooler. It is still very important to keep a cold keg cold. Never let your beer warm up unnecessarily.
Drink your creation. Notice how fresh your beer is. Notice how much better your fresh beer is compared with even top micro brews in your area. If not, try again. You'll get there.

[edit] Tips

Clean anything the beer sits in that is not boiling with OxyClean or your cleaner of choice. Fermenters should be cleaned and scrubbed thoroughly and then sanitized.
PBW and OxyClean are the same thing.
Five Star StarSan is the best sanitizer out there. Use 1/4 oz per 5 gallons. No rinsing. This stuff does not affect organic materials.
Don't get wrapped up in fancy equipment or nit picky methods. Concentrate on the overall basic process, keeping things clean and quality ingredients.
Brewing is fun and exciting and after a few brews under your belt brewing will be second nature.
An outdoor burner with a converted keg for boiling is a good investment. With a converted keg you can double your batch size to 10 gallons.
It is amazing how different your beer will turn out with different strains of yeast. Experiment with different stuff; you won't be disappointed.
Invest in soda kegs and build a kegerator, you'll probably have a lot more fun than messing with bottles.