Potions are magical liquids which give either a positive or negative effect for a period of time. There are two types of potions, Regular Potions which you drink (Default is Right Click) and Splash Potions which are thrown (Default is Left Click), effecting everything they hit. Brewing is the method to create both types of Potions.
You can find a complete list of potions Here.
To brew potions you will need to craft a Brewing Stand. To use a Brewing Stand, place it on the ground (Default is Right Click) and then activate it (Default is Right Click). The Brewing Stand has a single slot at the top for the material you want to add to the potions and 3 slots at the bottom for potions allowing you to make 3 potions per material.
You need to craft Glass Bottles to hold the potions you brew (See Here for the recipe). Once you have Glass Bottles you need to fill them with Water. With a Glass Bottle selected right click on Water to turn it into a Water Bottle.
Awkward Potions form the basis of most of the Potions. To brew them you need to place your Water Bottles in the bottom slots and Nether Wart Seeds in the top. Glowstone Dust, Redstone Dust, and Fermented Spider Eyes are the main potion ingredients. Brewing Gunpowder into a potion turns it into a Splash Potion.
You can find a complete list of potion recipes Here.
Enchantments give armor, weapons and tools special abilities. Enchanting requires experience levels which is gained by mining and by killing monsters. When your experience bar is full you will gain an experience level (Which will be displayed as a number on the experience bar). To enchant an item you will need to craft an Enchantment Table. To use a Enchantment Table, place it on the ground (Default is Right Click) and then activate it (Default is Right Click). Place your Armor, Weapon or Tool in the slot and 3 random enchantment options are displayed. If you have enough experience levels you will be able to select one of the 3 random enchantments.