Minecraft wiki entry for Brewing, associated text: Add a base ingredient to create a base potion, usually nether wart is used to create an awkward potion.
Add an effect ingredient into the awkward potion to create a potion with a working effect.
In any order, several optional steps can be added (depending on the basic effect, not all of these may be valid):
An enhancer ingredient may be added: glowstone dust to make the effect more intense, or redstone dust to make it last longer.
A fermented spider eye may be added to "corrupt" the effect, generally reversing it.
Gunpowder can be added to convert it to a splash potion, which can be thrown (or fired using a dispenser) to affect all players and mobs in a radius.
After a splash potion is brewed, Dragon's breath may be added to upgrade it to a lingering potion, which can be used to create a lasting area effect cloud, or to craft tipped arrows.