What is a data media device?
I'm writing this page because there seems to be a lot of confusion about how many of the newer Android devices work. Starting in Honeycomb 3.0 with the Xoom, Google changed the way that they handled storage. Instead of having a "data" partition with your apps and a separate "sdcard" partition for storage, Google started giving you a single, very large data partition. Inside /data is a folder at /data/media that contains all of the contents of what you think of as your internal sdcard.
Since /data/media is part of /data, we pretty much never actually format the data partition. Formatting data, of course, also removes the media folder that contains the internal sdcard. When you choose a factory reset, instead of formatting, we use rm -rf commands to remove all the folders except for the media folder so that we can remove all of your apps and settings while leaving your "sdcard" intact. In TWRP we also have a wipe internal storage option that rm -rf's the media folder and a "Format Data" option that formats to recreate the entire file system in case something goes completely wrong or to remove device encryption.
When you're booted to Android, Android fuses the media folder to /sdcard and emulates a FAT files system that doesn't have permissions for legacy apps. We don't currently have fuse in recovery, so we just add an extra mount command to mount /data/media to /sdcard so in recovery you still have to worry about permissions on /sdcard.
Because the "internal sdcard" is not a true FAT file system, you can't mount it via USB storage. Well, that's not technically true, but the vast majority of people use Windows computers and Windows doesn't recognize ext4. If we were to allow you to mount the data partition via USB storage, Windows would claim that the device wasn't formatted and offer to format it for you, which, as you can imagine, would be a disaster. The whole ext4 setup is another reason that Android switched to using MTP for transferring files. Most of these devices don't have the necessary kernel configuration to even support USB storage mode, so it's not very easy to enable USB storage if we even wanted to try. With TWRP 184.108.40.206 and higher, we now support MTP. MTP is the protocol most newer devices use to transfer files on your PC. If for some reason MTP is not working for you or if you're using an older version of TWRP, you can use adb to push and pull files to and from your device.
As a special note, if you choose to do a factory reset from your ROM, even if the ROM says that it will wipe everything including the internal storage, well, that's not what TWRP will do. A stock AOSP recovery would format data including the "sdcard" but TWRP will use its regular factory reset setup that leaves the internal storage intact.
There are a couple of nice gains with using this setup vs the old data + FAT storage partition. With /data/media you, as the user get more control over how you use your storage. If you have a ton of apps, then that's no problem since you have a huge data partition to work with. If you don't have a lot of apps, you get more room to use for storing things like movies. Further, ext4 doesn't suffer from the 4GB file size limit that FAT has, so you can have a large, high-def movie on your device if you like. I'm sure another motivating factor was to get Android away from using FAT which is a Microsoft creation. Performance on ext4 in Android is also probably better than FAT. As a downside, data media devices tend to store a lot more app data in the "data" section and so backups on these devices tend to be larger.
Android 4.2 has changed things with /data/media devices a little bit due to the multi-user support that came in 4.2. Each user is assigned a subfolder in /data/media. The main user gets /data/media/0 and subsequent users get /data/media/10 and /data/media/11 and so on. If you switch users in Android 4.2, the system will remount the /sdcard folder to point to the proper user's folder. TWRP has been updated to use the /data/media/0 folder starting in 220.127.116.11.
Another "feature" of 4.2 is that when you "update" to 4.2 it may attempt to upgrade your /data/media to multi-user. If you're running an older version of TWRP than 18.104.22.168 or newer, a factory reset may trigger multiple upgrades, causing your "sdcard" to get moved to /data/media/0 then /data/media/0/0 and then /data/media/0/0/0 and so on depending on how many times you "upgraded". This may cause backups to not be visible in TWRP. Also, there currently isn't a good way to go back to a 4.1 ROM after using a 4.2 ROM without having to manually move your files around.