Instructions
Download the ZIP file for a custom map that you want to use. Mindcraftforum.net contains possibly the largest collection of maps, so that site is a good place to start looking (see link in Resources). Note whether the map is a survival or creative map.
Double-click the ZIP file, and then click ̶0;Extract All.̶1; Choose a location to extract the files to, make sure the ̶0;Show extracted files when complete̶1; box is ticked, and then click ̶0;Extract.̶1;
Launch "Minecraft Pocket Edition" and create a new map. Call it anything you like, but make sure you select the same type -- survival or creative -- as the map you downloaded in Step 1. Exit the game when you've created the map.
Connect your device to a computer, and then transfer the three map files to the following location on your device's SD card, replacing ̶0;MyMap̶1; with the name of the map you just created:
/games/com.mojang/minecraftWorlds/MyMap
Launch the game again. You have now replaced the map you created in Step 3 with the map you downloaded, and you can run it like any other map.