You can buy most of the things you will need to make your own beer at your local home brewery store.  The most basic items you will need are a large boiling (or brew) pot, a tablespoon, a measuring cup, a glass jar, a fermentor (a big plastic food grade tub works, but it must have a tight lid), an airlock, sanitizer (you can use a chemical sanitizer or bleach), yeast (liquid yeast works well), and hops of your choice.

Pipe Heat Tape

Remembering to sanitize everything that touches your beer once it has been boiled is very important.  Keeping bacteria out of your brew will ensure your beer ends up tasting right. If you're using beer taps make sure that your use pipe heat tape to insulate the pipes.

There are three steps to making home brewed beer: brewing, fermenting and bottling.

For the brewing process you will activate your yeast and steep the hops you have selected, along with any malt for dark or spiced beers.  This will create wort.

Fermentation begins with removing any excess material using a sterilized strainer from your brew and aggressively pouring the liquid into the fermentor bucket.  Put the lid on and let it sit for about 2 weeks at room temperature to ferment.

Bottle your beer with clean, sterilized bottles.  All you need to do is pour the beer into the bottles and carbonate.  You can use carbonation tablets or a mixture of sugar and water to carbonate.  If you use the sugar method, you will need to cap the bottles and let them sit another two weeks to carbonate.  If you use carbonation tablets, just cap them and let them dissolve.