The first step in a partial grain brew (this is where some grain is used in addition to malt extracts) is to steep the grain in 160°F water. The pulls out the sugars and the flavors. You want a container that's relatively temperature stable, so I'm using a 2 gallon picnic cooler. This kills two birds with one stone, because later the sugars get washed out of the grain and you'd like to leave the husks behind. To this end I pulled the spigot from the cooler and replaced it with a stopper with a copper tube that has a bunch of slits cut into it. It makes a fine filter. Note the steam.
While all that is going on, it's handy to start some water boiling. Enter the next piece of essential equipment, a nice big stock pot. This is a really cheap one which burns a little because the metal is so thin, but it's 16 quarts and that's the perfect size for brewing. 2 gallons to boil.
This is the outside view of the cooler. Not too exciting, eh? There's a hose coming from the copper pipe and a hose clamp. Don't forget to latch the hose clamp before you put in the water. I've done that exactly once. Oops.
Once the grains have steeped long enough, rinse water goes through. This process is called "sparging" It sounds like something that a pirate would do. I set the cooler up on a height to make it a little easier to work with.
Here's the rinse water, now tinted with malt sugars.
Now, normally I don't recommend cracking a cold one while you're brewing (or at least not too soon), but this was waiting in the keg. My last beer was a stout modeled after Rogue Brewery's Shakespeare Stout. My local brew shop guessed at the recipe and did a terrific job. It's a pretty close match, although they think it could've used some more oats.

Here are the remaining ingredients/equipment. The big brown bag is malt extract. It takes a lot of sugar to make beer and to do it full grain, you need a lot of grain and a lot of boiling. This means a really, really big pot and stove. I have neither, so I don't do full-grain. The malt extract goes in first.

The white bag on top is Burton's Salts which is used to change the hardness of the water.Some recipes use gypsum, which is what's in drywall.

The greenish bags are hop pellets. One is for bitterness, and one is for aroma. They get added at certain times during the boil.

The tiny bag is Irish moss, which is a source of carageenan. This is an emulsifier which removes particulates making a clearer beer.

One of the handy things about brewing in the winter is that you can cool off the whole shebang pretty quickly. This bucket is where the fermenting will happen. This can't start until the wort (another brewing term) has cooled enough so that it won't kill the yeast.
My next door neighbor grew hops last year and gave me a lot of his harvest. Hops are usually used in a processed form which is convenient and predictable. This is where it all starts from. These are hop flowers.
And in they go. Putting them in at the end is called "dry-hopping".
Last but not least is an airlock to keep things nice and airtight during fermentation.