Guide to Minecraft Potions

Introduction
Much like enchantments, potions are an essential part of Minecraft, and give the player many utilities in the form of buffs, throwable area-of-effect damage, resistance to certain types of damage, and more.
Please keep in mind that potion durations are noted as minutes:seconds (e.g. 1 minute and 30 seconds is noted as 1:30, and 4 minutes is noted as 4:00).
How to Brew Potions
Potion brewing begins with a brewing stand (crafted using cobblestone and a blaze rod, and powered using blaze powder), water bottles, and netherwart.
First, obtain a brewing stand by either crafting one or stealing one from a village.
Next, you'll need to power the brewing stand (using the slot to the far left) using blaze powder. This gives 20 "charges" to the stand, and one charge is expended each time something is brewed.
The first potion you need to brew is the awkward potion. It has no effects on its own, but serves as the basis of (almost) all other potions. Do this by placing netherwart in the top slot and three water bottles in the bottom slots.
If you're making Weakness potions, you can skip this step! Weakness is the only potion in the game that can be made without an awkward potion. Just add a fermented spider eye to water bottles in the brewing stand.
Now that you have awkward potions, you can create whatever potion you need! Simply place the potion you're modifying in the bottom slot(s) of the brewing stand, and the ingredient you're using in the top slot (and make sure the stand is powered!).
Types of Potions
No Effect: Mundane, thick, and awkward potions have no effects. While mundane and thick potions can't be used for anything and are largely entirely useless, awkward potions are perhaps the most important potion: they serve as the base for almost all other potions. You can't just brew ingredients with water bottles to make potions (except in the case of Weakness potions); you need to brew them with awkward potions.
Enhanced: Most potions can be enhanced using glowstone. This makes the potion's effects stronger, but often reduces their duration. Enhanced potions cannot be extended.
Extended: Most potions can be extended using redstone. This makes the duration longer, but doesn't affect the potion's strength. Extended potions cannot be enhanced.
Splash: All potions can be made into splash potions using gunpowder, which can be thrown and which apply their affects to an area of effect (AOE) around where the potion landed. This form is most useful for potions with negative effects (like harming or poison) because the potions can then be thrown on mobs, but they're also helpful for applying positive potion effects (like slow falling, instant health, or regeneration) to the player very quickly when needed.