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What is the PEM?

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"I did not know that" (Johnny Carson)
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Power Electronics Module. It controls the power to, and regeneration from, the motor, converting DC from the battery to the AC needed for the induction motor. It also converts AC to DC to charge the battery. On the Model S these functions are separated again.
 
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Yes it is a big inverter, but it also has the charging circuits inside too.

In the early Roadsters (2008 1.5s) the PEM used an AC propulsion licensed "reductive charging" circuit that apparently used motor windings as part of the charging circuit. In the later cars (2.x 2010/2011/2012) they redesigned the PEM with what I recall were the following changes:
#1: Smaller
#2: Cheaper to produce
#3: Easier to service
#4: Redesigned charging that didn't use the ACP reductive charging license anymore.
#5: New Sport variant available.

There are also various discussions about merits and pitfalls of how the air cooling works in the different PEMs.

I think some of the very early (prototype) Roadsters had analog circuits in the PEMs that were later replaced by digital circuits, but the original analog design never saw production.