Steps

This will explain how to export worlds from your device onto your computer.

On Android

1. Plug in your device to a computer via USB
2. Open a file manager
3. Go to the device's folder
4. Open the "games" folder then the "com.mojang" folder under it
5. There you will find the "minecraftWorlds" folder, open it and your worlds will be there
6. Since the worlds have random names (e.g mA8AACAzBwA=), open each world's folder and open the "levelname.txt" file which will say the name of the world you opened
7. Once you find the world (or worlds) you want to export, copy and paste the folder to some place on your computer
8. If you want to import it to a new device just reverse the process (copy the world from the computer to the "minecraftWorlds" folder)

On iOS

1. Install iFunbox (I will tell you how to do this with version 4.0)
2. Plug in your device to the computer that has iFunbox installed
3. Launch iFunbox and open the "My Device" tab
4. Click on "Apps" and open the sandbox for Minecraft
5. Find the "minecraftWorlds" folder
6. Do steps 6-8 in the "On Android" section