Note that this applies only to the U.S. section of the ISS.
There are 3 different types of MDMs used on the ISS.
Tier 1: Top level control. These are inside the ISS and they have 3 on board, but only one is active.
Tier 2: There are 5 different types of these, only one is external to the ISS... it is called the EXT MDM. There are 2 of every tier 2 MDM, one primary and a backup.
Tier 3: I think there are 5 of these as well, these are sensor/component level control, there are over 30 of these in the ISS if I'm not mistaken, some are inside, some outside.
From a compatibility standpoint the MDMs come in 3 different sizes, which hold 4, 10, or 16 cards. Cards and the entire assembly is easy to move around and essentially slide in and out. My understanding is that they are generally specialized cards in each MDM and cannot be swapped for other good ones, hence the backups for each system.
The one that failed recently is EXT-2 MDM (Tier 2) with a size of 16 (not all are used). A spare EXT MDM is inside the ISS, but will have to be mounted outside before it would be useful.
This article is an excellent read regarding the failure of the EXT-2 MDM and discusses the options available to the flight crew.