When you have the required ingredients potions can be brewed and drunk to momentarily give you special abilities like faster running and night vision. The Crafting ingredients to brew these potions will mean entering The Nether to obtain them. Before you enter The Nether make sure you are kitted out in high powered armour and have a sword. The three types of potions that can be created are:
Splash Potions
These potions are thown and will affect any players/Mobs within range of the splash. The bottle will break upon hitting any Mob or Block and give off particles.
Drinkable Potions
These are drinking potions that when drunk give a status effect for a certain duration of time. It will not affect any Mob/Player around the player, and the player will retain the empty bottle for further use.
Lingering Potions
These potions must be thrown like a Splash potion. They will create a 'cloud' of that status effect in a radius which will last for roughly 30 seconds.
Brewing Stand
You will need to create a brewing stand at a Crafting table for your potions. This is easily achieved by using three cobblestone blocks to go with your Blaze Rod.
Base Potions
Water Bottle
Use: Base for ALL Potions
How to make: Glass Bottle + Water
Awkward Potions
Use: Base for primary Potions
How to make: Water Bottle + Nether Wart
Thick Potion
Use: Base for Potion of Weakness
How to make: Water Bottle + Glowstone Dust
Mundane Potion
Use: Base for Potion of Weakness
How to make: Water Bottle + Any Ingredient
Long Mundane Potion
Use: Base for Potion of Weakness
How to make: Water Bottle + Any Ingredient
Types of Potions
Potion of Instant Health
Effect: Instantly restores health
How to make: Awkward Potion + Glistering Melon
Potion of Swiftness
Effect: Increases Movement Speed
How to make: Awkward Potion + Sugar
Potion of Slowness
Effect: Lowers speed of Mob/Player
How to make: Potion Of Swiftness + Fermented Spider Eye
Potion of Regeneration
Effect: Restores health over a period of time
How to make: Awkward Potion + Ghast Tear
Potion of Harming
Effect: Does instant damage
How to make: Potion of Poison + Fermented Spider Eye
Potion of Strength
Effect: Increases attack damage
How to make: Awkward Potion + Blaze Powder
Potion of Weakness
Effect: Lowers attack power
How to make: Water Bottle + Fermented Spider Eye
Potion of Fire Resistance
Effect: Gives immunity to Fire and Lava
How to make: Awkward Potion + Magma Cream
Potion of Night Vision
Effect: Gives ability to see better in dark areas and underwater
How to make: Awkward Potion + Golden Carrot
Potion of Underwater Breathing
Effect: Gives ability to breathe underwater
How to make: Awkward Potion + Pufferfish
Potion of Poison
Effect: Degenerates health over time
How to make: Awkward Potion + Spider Eye
Potion of Leaping
Effect: Gives a jump boost of 1 1/2 Blocks and the ability to jump over fences
How to make: Awkward Potion + Rabbit's Foot
Potion of Invisibility
Effect: Grants Invisibility
How to make: Potion of Night Vision + Fermented Spider Eye
Potion of Decay (Exclusive to Minecraft PE)
Effect: Grants the Wither Effect
How to make: Unknown
Modifiers
Redstone
Effect: Increases duration of the Potion
Glowstone
Effect: Enhances the Potion's effects
Gunpowder
Effect: Converts a normal Potion into the splash variety
Fermented Spider Eye
Effect: Corrupts the Potion effects (only in certain cases)
Dragon's Breath
Effect: Used to convert a Splash Potion into a Lingering Potion
The video below is a tutorial showing you how to make potions.