like Ryccardo said, wad is a container used as "installation package".
you can see them like a .zip or a .rar with all the game's files inside.
There are 2 different methods install and play a game in wad format :
Note that installing the WAD means that the "wad manager" read the .wad file, and extract its content to your console. Once installed, the game appears as "channel" on your Wii system menu, and you launch the game as if you bought it from eShop.
1- install the WAD to your console's internal memory. It's the best compatibility method, but your Wii has limited free space, and is limited to 48 games at the same time.
2- install the WAD to your console's official SD menu. It's the second best, nintendo added a way to extend your console's memory by installing the game to SD card. But you can't launch it from SD. to launch the game, the console need enough free space on its internal memory to copy the game from SD to internal, then launch it. If you play another game, it deletes the "temporary copy" located on your console to make space.
These are using the official console's interface to play the games.
There are another method called "emuNAND", which is using NAND redirection feature.
If you don't know what it is : You can make a copy of your console's file system to your SD card or USB, and redirect all file access to it. the console uses that copy as if it was the real internal memory (called NAND), but it's actually accessing the redirected NAND copy instead.
The advantage is that you are not limited to the console's size anymore, you have up to 16TB of free space to install all your games ! (instead of 512MB)
The second advantage is that this is region free ! you can install more games.
There are 3 emuNAND mode :
- cIOS emuNAND : this is the easiest and most common mode. it's a simple copy, and the cIOS is responsible for redirecting NAND Access to emuNAND folder. Expect 80% game compatibility.
- Neek emuNAND : this one requires a little more steps to setup, but once done it's sharing the same NAND copy than "cIOS emuNAND" so you don't need to setup 2 different copy. the entire console is redirected to the copy, not just the game. Expect 99% game compatibility.
- Hermes cIOS emuNAND : very old, nobody do that anymore. It could play wad file directly from the SD, without installation. I never tried. And I guess nobody did. please, don't ask, you will not get help anyway.
emuNAND guide I wrote :
https://gbatemp.net/posts/6592730