I have a world on Minecraft that I've had for the past two and a half years. I've made a lot of progress on it, and it has been on survival mode the entire time it has existed.
However, the computer that it is on is five years old, and I feel it's near it's final days. I really don't want to lose my Minecraft world. There's too much work and nostalgia I have from it to let a dead computer take it away from me.
How can I transfer this world to another computer?
You could copy the save file to an external storage device and add it back onto the new computer.
I believe you can find and copy the file by searching & running %appdata%
Copy the files of your minecraft folder related to the world save through USB or external hard drive and transfer it to the new computer when you re-download minecraft on it.
Oh dear, almost every Minecrafter has heard of adding maps into the game. Makes the whole barrel of fun into a shipping crate!
Assuming you have Windows, Press [Windows] + R, and type in the %appdata% like the bloke said. Make your way to the Minecraft file saves and you'll have your world.
You can make backup copies if you desire, and once you've moved on just place it back where it should be on the new one.