Excuse my pedantic nature, but the correct terminology is Supercharger Cabinet, or Charging Cabinet. Inverters convert DC to AC. The Supercharger Cabinets convert AC to DC in a very controlled way to charge the Tesla.
The big green box inside the enclosure is the Distribution Center (a giant circuit breaker cabinet) or in Tesla speak, it's also a "Switchgear Assembly", and the moderate sized green box outside the enclosure is the Utility Transformer. The charging Spaces are also called Charging Stalls, and the Charging Pedestals are also called Charging Stations. Here is an annotated picture from Blanding, UT:
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If you want to look at the gory details of what goes inside and the plans for a Supercharger Site, see
http://www.cityofmadison.com/planning/projects/reports/89etm_site2.pdf