The MPD is the maintenance roadmap you might say, for the basic Maintenance Program agreed between the OEM and the regulator, that defines the inspection and test requirements to maintain airworthiness. It'll include a listing of which inspections and tests have to be done at Routine, A Check, C Check or some time interval, etc.
Inspections means things like zonal inspections and detailed visual inspections, that sort of thing.
"Tests" is generally Operational Tests (does the system work or not) and Functional Tests (does the system meet a performance standard - Under MSG-3, you can run systems indefinitely as long as they meet some performance or measurement limit). Inspections are generally Detailed Visual, close inspection of specific parts or a system, or Zonal, general inspection of a defined area.
The procedures you use on the floor will be Task Cards, that have their own manual that has all of the Maintenance Program test and inspection procedures, with a reference numbering system tied to the MPD.
The AMM is much broader, and is the technical source for the Task Card content and all other procedures on how to work on the plane, like Removal and Installation procedures, Return to Service tests, etc. When you peruse the AMM you'll find that some procedures have Task Card and MPD references at the start, and some are stand alone procedures with no external references (mostly Removal and Installation).
Task Cards will typically be an AMM inspection, ops test, functional test, or special test, as required by the approved maintenance program, and more or less extracted word-for-word from the AMM, so that when you are assigned a scheduled maintenance task you just need to be provided the Task Card and work from that.
The procedure you follow will be a verbatim extraction from an AMM procedure which is the foundational technical source for most maintenance on the airplane.
So, if you have to go beyond the content of the Task Card because you have to replace something that failed a test or inspection, or if you are doing something to the airplane not directly related to the maintenance program, like resolving snags, or doing an ops test or functional test for troubleshooting purposes, that's when you have to dig out the AMM procedure itself.