What is it called when a Drupal Module installs with several modules?
what is each piece called?


I don't have an official reference, but "submodules" or "sub-modules" is by far the most common phrase I've seen used for modules included in a main module to complement/extend its functionality. Prominent members of the Drupal community seem comfortable with referring to them as such.


  If they show up in one group, does that mean they are all written by the same entity?


It can in some cases, but there's nothing to guarantee it. Any module can list itself under any of those groups by setting the package key in its .info file. A lot of modules that provide field types list themselves under the generic "Fields" group, for example.