Potion Tips

To brew potions, you'll need a Brewing Stand, and to make a Brewing Stand, you'll need to be tough enough to venture into the Nether and beat up a few of the new Blaze mobs. Not familiar with Blazes? They're toughened clusters of magma and ash which fly and spit unblockable fireballs. Not exactly the kind of thing you want to bump into while you're exploring the inside of a volcano. Pro tip: snowballs are surprisingly effective against Blazes, so make sure you scoop up a few before heading through the portal.

Blazes drop Blaze Rods, and one Blaze Rod plus three Cobblestone blocks equals a Brewing Stand. To make a potion, you'll need three Water Bottles (made by combining three glass blocks, then waggling the results at a water source) or existing potions plus an ingredient. Mixing in Nether Wart produces the Awkward Potion, which has no effect by itself, but serves as a basis for all other potions. You can also brew up Mundane and Thick potions from water, but the range of potion types they provide access to is comparatively limited.
It's possible to craft scores of different potions by experimenting with item formulae. Once you've concocted a potion, you can optimise the effects further with modifying ingredients. Glowstone Dust increases the potency of the potion, Redstone increases the duration of its effects, and Fermented Spider Eye is used to reverse the potion's effect (e.g. "Swiftness" becomes "Slowness").
There's a handy diagram of potion combinations over at the Minecraft wiki.