Essential Equipment
First we'll go over the hardware you'll need in order to brew. Everything you need can be purchased at your local homebrewing store as a kit, or you can assemble most of it yourself, if you'd rather. Everything will be described in enough detail so that you should be able to figure out how to fashion all the equipment you'll need.
Fermenter – An airtight vessel with a hole where an airlock can be inserted. Typical fermenters are either glass bottles (bottled-water bottles are good for 5 gallon batches and are also known as carboys. 1-gallon jugs, of the type apple cider is typically sold in, are good for smaller batches,) or large plastic buckets with tight-fitting lids. The important thing to remember is that it needs to keep any wild yeast or bacteria that are floating around the atmosphere out of the brew you're making.
Airlock – Seals the fermenter and allows the gases produced during fermentation to escape. Relatively cheap airlocks (with a stopper to fit your fermenter) can be purchased at your local homebrew store, or you can make one yourself using a piece of plastic tubing and another jar filled with water.
Figs. 1 & 2: Types of airlocks
Bottles – For a 5 gallon batch, 3 cases of returnable beer bottles is enough. (Non-returnables won't stand up to repeated handling and you'll end up with broken bottles.) You can also use champagne bottles (you'll need 30 or so). Grolsch-style bottles work well too, and don't require a bottle capper. Some people have successfully used plastic 2-liter pop bottles, but the plastic will let some gases through, and the screw-on caps don't seal very well after the first time. For those reasons, I wouldn't recommend going that route.
Bottle-capper – This can be bought at a homebrew store near you.
Bottle caps – Gotta have something to keep the mead in the bottles.
Brewing pot – If you're going to be making a mead with fruits or spices in it, you'll want a pot to boil stuff in. I use a 3-gallon stew-pot, which works well for 5-gallon batches. You won't need anything much larger than 3 gallons initially. If you're not using fruits or spices, this'll still be handy for mixing things in.
Funnel – One with a filter or screen built into it is best, but any kitchen funnel will do. You'll be pouring into this from your brewpot, so that should give you an idea of how big it should be.
Tubing – You'll also want a supply of plastic tubing for transferring liquids about. I'd recommend having a couple 3-to-5 foot lengths of plastic tubing (if you discover you've bought a piece that's too long, cutting it shorter is easy. Making it longer, on the other hand, is a real bitch). One piece should be the same size as the hole in the stopper you're using in your fermenter (⅜” outside diameter), and the other piece should be larger (½” inside diameter or so) for siphoning the mead from the fermenter into bottles. If you're not in the US, the tubing to fit in the stopper hole will be about 9mm, and the other piece can be about 12mm. Best is to take a stopper with you when you go shopping.