Combat (1.8 and below)
Alchemy (Potion Brewing)
Combat (1.9+)
Annihilation
Need help with one of Minecraft's most popular mechanics? No problem!
DISCLAIMER: MY APOLOGIES IF PICTURES DO NOT WORK. JUST RIGHT CLICK WHAT YOU SEE AND SELECT "OPEN IMAGE IN NEW TAB" :D
REDSTONE

What is redstone?
Redstone (the general term, not the item) is Minecraft's equivalent of real life wiring and electronics. Though Minecraft does twist and change the functions of some blocks. Take dispensers for example, the real life equivalent of dispensers almost always dispense ice and drinks, while Minecraft dispensers dispense any item, even activating some, such as TNT and Spawn Eggs.

IMPORTANT MECHANISMS

Meet (most) your arsenal!

Redstone (AKA Redstone Dust, or Redstone Wire)
Redstone dust can only be placed on opaque blocks as well as glowstone, upside-down slabs and upside-down stairs, and hoppers.Redstone dust will configure itself to point towards adjacent redstone power components (such as levers and buttons) and transmission connection (like comparators and repeaters) points. Redstone dust will also configure itself to point towards adjacent redstone dust one block higher or lower – unless there is a solid opaque block above the lower redstone dust, or if the higher redstone dust is on a transparent block such as glowstone, an upside-down slab, stair, or a hopper.If there is only one such adjacent redstone component, redstone dust will configure itself into a line pointing both at the neighbor and away from it. If there are two or more such adjacent redstone components, redstone dust will point at them, forming a line, an "elbow", a "T", or a cross. If there are no such adjacent redstone components, redstone dust will configure itself into a directionless "dot", which can provide power in all four directions.Redstone dust will not automatically configure itself to point towards adjacent blocks. If it is required, the other neighbors of the redstone dust must be arranged to create it.

Piston
Pistons are blocks that can push the blocks they're facing at. The pushing strength has a maximum of 12 blocks. Otherwise the piston will simply not work.
Fill the top 3 with any kind of planks.
Fill the right most and left most mid and bottom slots with cobblestone.
Redstone in the middle.
Iron ingot on the bottom middle.

Sticky Piston
Same as a piston, but it can pull back the block it is pushing.
Place a slimeball on top of a piston.

Lever
Upon activation, will power any block or redstone-based block adjacent to it until it is flipped back.
Place a stick on top of a cobblestone.

Button (Wooden and Stone)
Wooden buttons power any adjacent block or redstone-based block for 1.5 seconds. Can be activated by arrows, keeping them pressed.
A single wooden plank on any slot.
Stone buttons power any adjacent block or redstone-based block for 1 second. Can only be interacted with by players.
A single piece of smooth stone on any slot.

Tripwire Hook
By linking two of these with a line of string, any player, item, or entity that steps on that line will make the hooks produce a signal.
Iron ingot on top.
Stick in the middle.
Any wooden plank at the bottom.
Must be aligned.

Redstone Block
This block powers stays permanently on, and will power any adjacent block until it is moved out of the area.
Fill the crafting interface with redstone dust, similar to iron blocks and diamond blocks.

Iron Door
Just like a regular door, but requires it to be powered before opening.
Craft a door, but with iron ingots.

Redstone Repeater
This block "repeats" a redstone signal. A regular redstone signal lasts 15 blocks long. So adding a repeater allows the signal to go 15 more blocks.
This block also has a "locked" state, that occurs when it is powered from the side (not directly, but from other repeaters). In this state, the repeater's second torch will rotate and become bedrock, preventing it from "repeating" the signal unless the power from the side is removed.
Fill the bottom with smooth stone.
Place two redstone torches on the sides on top of the smooth stone.
Place redstone dust in the middle.

Redstone Torch
Similar to a redstone block, redstone torches will power anything adjacent to them, with the exception of the bottom block.
Redstone torches can be turned off by powering the block they are placed on.

Dispenser
These are blocks that when powered, dispense any item inside of it. It can also activate some items, such as spawning in primed tnt instead of dispensing out the block, and creating mobs instead of dropping the actual spawn egg.

Daylight Sensor
This block inputs signal to any redstone-related device next to it, with the signal strength based on the time of day. Getting weaker the closer it is to night.
Fill the top with glass.
Fill the middle with quartz.
Fill the bottom with wooden planks.

Redstone Comparator
This block "compares" a signal, or in a simpler way; detects the status of certain blocks, and converts them into redstone signal, powering any block in front of the "triangle". So if you place a comparator (facing away) next to an item frame, it will output a redstone signal based on the item in the item frame's rotation. Do it with a jukebox, and it will output signal based on what disc is inserted.

BASIC MECHANISMS

Torch Elevator
This elevator, made using redstone torches, allow a signal to travel upwards.
So, in this picture, the piston is off. And the button is not connected to the piston.

But, If I hit the button;

The button will turn the torch (which was turning off the torch above it, which was connected to the piston) off, and allow the torch above it to power the piston.

Torch Elevator (Downwards)
This is similar to the torch elevator mentioned above, but it functions to carry the signal downwards.
So, in this picture, the piston is on because of the redstone torch;

But, if I hit the button, the button turns of the torch that was turning of another torch, thus allowing that torch to turn off the torch powering the piston (My apologies if this doesn't make sense xD).


Clock
This device's signal "pulses", or repeatedly turns on and off.
So in this picture, nothing is really happening, now is it?

But I turn this on...

The dispenser is repeatedly firing arrows because of the pulsing signal.

LOGIC GATES (BASIC)
Logic gates are basically a series of inputs connected together to make a very specific output. Logic gates are commonly combined with other logic gates to produce even more specific outputs.

AND Gate
The AND gate, like the name implies, requires both inputs to be on in order to produce a signal, so...

The two inputs are the redstone torches, and because they are off, the piston will not extend, but...

If we leave the torches on, the signal will allow the piston to extend.

OR Gate
The OR gate, like the name implies, will produce a signal as long as one of the 2 or more inputs are on.
So if I flick the right lever, the piston will retract. But...

If I flick the other level, it will do the same thing.


Inverter
Allows the conditions for a logic gate to produce the opposite outcome. Creating other logic gates, such as;

NAND Gate
The opposite of the AND gate, this gate requires both inputs to be off in order for the signal to be produced.





Alright, I'm done, this took FIVE hours to do.
Thanks for reading! Leave any suggestions! :D