Hops - There are dozens of hop varietals, with widely ranging flavors. Wikipedia has a pretty good list.
Malt extract - Available in liquid (LME) or dry (DME) forms, this extract is a concentration of the sugars present in the wort after the mash. Using extract obviates the need for a mash. So far all of our beers have been extract based, with specialty grains for flavor.
Base malt - Malt that has been relatively lightly kilned, so as to not destroy the starch-converting enzymes. Base malt is the only malt that must be included in beer, since only base malt can produce fermentable sugars. Common varieties of base malt include: Pilsner, 2 row, Munich, and Vienna.
Crystal; crystal malt - Malted barley that has been stewed and then dried, in a process that causes starches to be converted into sugars via enzymatic action. These malts do not need to be mashed (i.e., steeped carefully to allow this very enzymatic conversion), and can be steeped very casually. Crystal malts add caramel-type flavors and unfermentable sugars to a beer, making it sweeter and more viscous. Crystal malt is categorized according to how dark it has been roasted, where a higher number corresponds to a darker malt and a more intense caramel flavor.
Roasted barley; black barley - Unmalted barley that has been very deeply roasted. Gives beer a nutty, coffee-like flavor. The characteristic flavor of modern stouts.
Black patent - Malted barley that has been very deeply roasted. Similar in flavor to roasted barley, but less coffee-like and with a bit more of an smoky edge. Often included in smaller quantities than roasted barley in stouts and porters.
Chocolate malt - Malted barley that has been roasted slightly less than black patent. Flavors of unsweetened chocolate and bread-crust dominate. The characteristic flavor of modern brown ales.
Original gravity (OG) - A measurement of the specific gravity (i.e., density) of the wort before fermentation. Water has a specific gravity of 1, and a liquid with anything held in solution will increase that liquid's specific gravity. Therefore, a wort with a high specific gravity, such as 1.080, is a wort with a lot of stuff in solution, which are usually fermentable sugars.
Final Gravity (FG) - A measurement of the specific gravity (i.e., density) of a beer after fermentation. As the sugars, which are heavier than water, are converted to alcohol, the density of the beer decreases. A beer with a low final gravity is a beer with very few unfermented sugars left—a dry beer, in other words.
ABV - We calculate alcohol content by subtracting a beer's final gravity from its original gravity and multiplying by a nonlinear coefficient, which is about 131 in the usual range for beer. For example, the Golden Bear has an original gravity of 1.073 and a final gravity of 1.008.
1.073 - 1.008 = 0.065
0.065 * 131 = 8.5
Thus, the Golden Bear is a calculated 8.5 percent alcohol.
International Bitterness Units (IBUs) - A measure of the concentration of isomerized alpha acids, which are the main bittering compounds in beer. A beer with higher IBUs will probably taste more bitter. We calculate IBUs using the Tinseth formula, which we've found to be closer to what we perceive in our beers than the Rager formula, which is the other one commonly used. In any case, calculated IBUs of any sort should be considered only an estimate.
Alpha Acid Percentage (AA%) - A measure of the concentration of alpha acids in hops. Hops high in alpha acids have a high bittering potential.
Acetaldehyde - A yeast byproduct with a sharp, green apple flavor. Generally considered an off-flavor at detectable levels.
Diacetyl - A yeast byproduct with a buttery flavor. Generally considered an off flavor at detectable levels.
Esters - Usually yeast byproducts; typically associated with fruit flavors. English ale yeasts produce moderate amounts of esters. Hefeweizen yeasts and some Belgian ale yeasts produce high levels of esters.
Fusel alcohols - In addition to ethanol, yeast produce smaller quantities of heavier alcohols. Fusel alcohols contribute boozy flavors associated with wine and spirits, which can be sharp and unpleasant.
Phenols - Usually yeast byproducts; typically associated with spices such as clove or black pepper. Hefeweizen yeasts and some Belgian ale yeasts produce high levels of esters.
For information on beer styles, see the BJCP style guidelines.
Mash tun: An insulated container that maintains a near-constant temperature over the course of a mash. Our mash tun is as yet, incomplete, due to lack of the proper plumbing pieces and lack of desire to drive to Shoreline to get said pieces.
Brew kettle: A large container that holds the wort during the boil. Ours is a big (7 gallon) stainless steel pot that we put on our stove.
Our brew kettle with wort chiller
Wort chiller: Not strictly necessary, but very helpful. We've recently purchased a copper immersion wort chiller, which replaces our previous system of putting the brew kettle in the bathtub with a bunch of ice cubes and/or snow, frozen 2L soda bottles, &c.
Immersion wort chiller
Fermenter: The vessel in which the beer is fermented. We've used 7-gallon food-grade plastic buckets, a 6-gallon PET carboy (a container shaped like the water jug on an office cooler), and various 1-gallon glass wine and cider jugs.
The fermentation chamber
Serving/carbonation system: Since we bottle condition all of our beers, our bottling system serves as both a carbonation and serving method. We have a bucket with a spigot that we use to decant the beer into (from the fermenter), as well as mixing the priming sugar into the beer. We then attach a bottling tube to the bucket's spigot and draw the beer into bottles, capping them with our hand-capper. Three weeks later, the beer is carbonated and ready to drink, though pouring the beer off the yeast and into a glass is recommended. Keep the bottle upright and pour slowly so as not to disturb the yeast that have settled to the bottom. Stop pouring when you see the yeast start to reach the mouth of the bottle, leaving about 1/4 - 1/8th of an inch of beer left in the bottle.
If you're serious about brewing, check out John Palmer's How to Brew, which is one of the best introductory texts that I've read on any subject.