Introduction
Every minecraft world has spawn chunks. This area is very important for certain farms, redstone machines, and other game mechanics, as well as for late-game gameplay.
Spawn Chunks
Minecraft worlds generate in "chunks" -- 16x16 block pillars that load in one at a time as the player explores the world. Chunks are very important for things like world and ore generation, biome blending and borders, mob spawning, and other things that the player doesn't generally need to worry about (though we'll come back to the mob spawning detail later).
When the player moves around the world, they're loading in a number of chunks dependent on their render distance. For our purposes in this article, chunks can have three different "states."
Ungenerated chunks, ones that the player has not yet rendered;
Generated and unloaded chunks, ones that have been generated but which the game "times out" and unloads to minimize resource use, in which game mechanics such as mob spawning will not function; and
Generated and loaded chunks, ones that have been generated and are "loaded," meaning that game mechanics such as mob spawning, redstone machinery, crop growth, etc. function normally.
The world's spawn chunks are a 19x19-chunk area centered on the world spawn point. These chunks will always be loaded, even if the player is not in or near them, meaning that certain game mechanics (below) will function as if the player were there. If the player enters the Nether or the End, the entire Overworld dimension will "timeout" and become inactive, so the spawn chunks will be unloaded and will not function.
In the 19x19 spawn chunks:
Iron farms will function as normal.
Baby passive mobs will continue to grow.
Redstone machines will function as normal.
Passive entities will not despawn and will contribute to the passive mob cap.
Hostile entities that are not custom mobs (see below) will despawn when the player moves 128+ blocks away.
Blocks that have progress bars (furnaces, blast furnaces, smokers, and brewing stands) will continue to work, including automated systems.
Crops and trees will not grow.
No mobs (except iron golems) will spawn.
The spawn chunks are important for certain farms, redstone machines, and other game functions. But perhaps the most important thing about them is that they contain the world spawn point. In post-dragon fight gameplay, the player will be moving between the End and the Overworld many times; but if the player broke the last bed they slept in prior to entering the End, the portal to the Overworld will place them at the world spawn point. As such, it's very important that the spawn chunks are always safe, and it's not a bad idea to have some kind of base there (if not your main one).
Uses for Spawn Chunks
Iron Farms
This is probably the most useful thing that can be put in your spawn chunks, and if you ever build one of these, this is where it should be. Because iron golem spawning uses a different mechanic than normal mob spawning, golems will still spawn while the player is away from the spawn chunks, so the iron farm will produce iron whenever the player is in the Overworld regardless of how far away from the chunks they are. While normal mob spawning won't work (hostile, neutral, and passive mobs won't spawn unless the player is nearby), golem spawning will, making it a great use for this game mechanic.
Automatic Brewing
Brewing stands in the spawn chunks will continue to progress even if the player is away, and redstone machines will continue to work, so an automatic brewing setup is a great thing to put here if you need a lot of potions.
Automatic Smelting
Similarly to the brewing setup, having a furnace (or smoker, or blast furnace) array running in the spawn chunks is an excellent idea. It will continue to smelt things even while you're away, so you can smelt a lot of items regardless of where you are.
Redstone
Redstone machines will continue to work in the spawn chunks. This means that fully automatic brewing and smelting arrays will work, as will item transport systems, redstone sorters, redstone doors and other similar machines.
On Servers
On multiplayer servers, it's best to be careful about what you or a Minecraft build team put in the spawn chunks. Having too many entities (mobs, item frames, dropped items, etc) loaded in the spawn chunks at a time can lag the server, as can having too many complex redstone machines running there.
In conclusion, Minecraft's spawn chunk mechanic is very useful for things like iron farms and smelting arrays, but it's also essential to ensure that the area is safe in case you break your bed and then travel through the End portal. Like the rest of Minecraft, it is what you make of it, so the potential really is limitless.