As discussed on the Mods and Addons page, the official Zan's MiniMap plugin is not allowed due to it showing caves. However it is still a very good MiniMap, so oliverw92 has created a modified version which does not contain the cave view mode.
You will need something that can open and edit .jar files - on Windows you can use 7-Zip.
Navigate to your Minecraft.jar. This is NOT your Minecraft.exe. On Windows, this is located in %appdata%/.minecraft/bin. Simply type this into a Windows Explorer bar to go to it.
Right click the Minecraft.jar and open it with 7-Zip
Open up the .zip file you downloaded.
Copy all the files from the 'modloader' folder into the Minecraft.jar. Literally copy-paste them into WinRar. Skip this step if you have modloader installed already
Copy all the files from the 'ZanMiniMap' folder into the Minecraft.jar, same as above.
Delete the 'META-INF' folder from Minecraft.jar in 7-Zip
Close 7-Zip
Start Minecraft - there should now be a lovely MiniMap available for you!