I've heard of level A, B, C, and D simulators, (as well as FTD and AATDs) but know nothing about them.
What criteria is used to certify a sim as a particular level?
Here there is a nice tabular representation of the requirements for each level.
Wiki provides a handy summary:
Full Flight Simulators (FFS)
FAA FFS Level A - A motion system is required with at least three degrees of freedom. Airplanes only.
FAA FFS Level B - Requires three axis motion and a higher-fidelity aerodynamic model than does Level A. The lowest level of helicopter flight simulator.
FAA FFS Level C - Requires a motion platform with all six degrees of freedom. Also lower transport delay (latency) over levels A & B. The visual system must have an outside-world horizontal field of view of at least 75 degrees for each pilot.
FAA FFS Level D - The highest level of FFS qualification currently available. Requirements are for Level C with additions. The motion platform must have all six degrees of freedom, and the visual system must have an outside-world horizontal field of view of at least 150 degrees, with a Collimated (distant focus) display. Realistic sounds in the cockpit are required, as well as a number of special motion and visual effects.
It's a complicated system based on the past. It started with FAA classification of Full Flight simulators (FFS), with visual and motion, and Flight Training Devices (FTD) without visual and motion. The letter system A,B,C,D is still used by FAA and EASA, the number system is replaced with acronyms such as MCC and FNPTII.
FFS for fixed wing:
FTD for fixed wing: old system
FTD new system:
Helicopters follow grosso modo the same requirements. A Level D has a huge field of view to enable to look down and see the runway for landing. For helicopters Level B is an attractive option: full motion and visual allows you to train in the dark art of hell hovering. Main difference with Level D is in the data collection: instruments can be read from a video instead of having to be downloaded from the aircraft internal data bus and measured with external transducers.
You will find a perfect description for each kind of simulator in the FAA Part 60 Change 2, the EASA CS-FSTD(a) 2012, the ICAO 9625 Edition 4, the CCAR 60 and many others.
Some authorities based their need on EASA or FAA or ICAO documents like CAAS (Singapore) is using the ICAO document.
For example, EASA is describing each level of Flight Training Devices in that table. And you will see a checkmark when the device will need to respect that requirement. And a small description for each level:
The goal behind that is to be able to fulfill the training requirement for pilots. And on which kind of device we can train the pilot as a necessary level to the license needs.
FAA Part 60 Change 2: https://www.faa.gov/about/initiatives/nsp/media/14CFR60_Searchable_Version.pdf
EASA CS-FSDT(A) 2012: https://www.easa.europa.eu/sites/default/files/dfu/CS-FSTD(A)%20Initial%20Issue.pdf
ICAO Edition 4: Not Available Freely on Internet