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Ok, now you have your equipment, and ingredients for your batch of beer. Remember that this article (and site) always assumes a 5 gallon batch of beer. If you are using a beer brewing kit that assumes less than 5 gallons, you'll need to adjust accordingly.
First thing is the water. Fill your kettle, pot, or whatever it is you are going to start the process in to about 2/3 full. Place the kettle on the biggest burner on your stove, and crank it up, the more heat the better. After you see bubbles in the water (it's starting to boil), add your malt extract. You'll need to stir and stir for a bit to get the think syrupy gook that is malt extract to mix with your water. After it seems to be mixed together well, you can stop stirring for a while. The concoction you now have on the stove is called "wort" (pronounced 'wert')
The primary purpose of this boil is to get rid of any living organisms that could turn your new awesome beer in to "yuch in a bottle". Almost any home brew beer will taste great, unless it's contaminated by micro-yuchies (more scientifically known as bacteria).
Remember, not even nasty funky "yuch in a bottle" can hurt you, but it can make a batch of beer have flavors that you don't want. More accurately, no bacteria can live in a bottle of alcoholic liquid. But, during the fermentation process, those bacteria, before they die, can make your beer taste poor. So make sure you boil your water/extract for a full hour. While you are waiting for that boil, you should take the time to sanitize everything.
SANITIZE you say??? Oh yah, if there is one part of brewing beer, be it in your closet or at Anheiser-Busch that is more important than any other step, it would be sanitation. Nothing can mess up a great batch of beer faster than little micro-beasties that funk up your beer. Cleaniness=Godliness is never more true than in the act of brewing beer.
It's time to sanitize everything that will come in contact with the wort that you just created. This includes the fermenter, airlock, strainer, and everything else that will contact the beer from now on.
It's really easy though. Simply mix about 2 tablespoons of normal, non-scented household bleach with each gallon of water that your sink holds. Dip and submerge every item that you plan to use into this solution for a minute or two. Soak the head of your cold water faucet in this solution (using a bucket preferable) numerous times. Now fill your fermenter (5 gallon food grade plastic bucket or glass carboy) about 1/2 with cold clean water and add 2 or 3 tablespoons of bleach. Slosh the solution around int he fermenter for a couple minutes. Then rinse the fermenter very thoroughly.
After all items have been sanitized, you can continue.
Fill your fermenter with enough water that when added with the wort that is billing will bring it to about 4 1/2 gallons (5 gallons if you have a 6 gallon fermenter). Use VERY cold water from the sanitized faucet. Now take the wort and using a funnel if necessary, pour it into the fermenter with the cold water.
Using a sanitized thermometer (Candy thermometer's work great), monitor the wort until it's temperature is around room temperature. Preferably about 70 degrees. Once you have reached that temp, shake the fermenter (or stir with a SANITIZED spoon) to thoroughly mix the solution, and get as much oxygen as possible into the wort. Add your yeast (just poor it in if it's dry yeast, more advanced yeast require preperation).
Place the airlock on the fermenter following the instructions that came with the airlock. Place the fermenter in a cool DARK place and wait. How long depends on the type of beer you are making, but usually falls between 48 hours and 7 days.
You can best tell when it is time to bottle your beer by watching the airlock. When it bubbles less than 1 time every two minutes or so, it's time to bottle the beer.