Which document class is being used?

You want the LaTeX kernel function \@ifclassloaded:

\makeatletter%
\@ifclassloaded{<someclass>}%
  {<true code>}%
  {<false coode>}%
\makeatother%

If your package executes \@getclass then the current class name will be stored in \@currentclass which you can then test or print out as required (this is probably more general than using \@ifclassloaded which requires a class name in advance). You can define \@getclass like so

\def\@getcl@ss#1.cls#2\relax{\def\@currentclass{#1}}
\def\@getclass{\expandafter\@getcl@ss\@filelist\relax}

and use like:

\@getclass
\typeout{this document uses \@currentclass\space class}

Actually a class is loaded with the same internal macros like packages (both are officially "files with options") and LaTeX doesn't store the class file name in a specific macro. So you can't directly test which class was used, only test whether a specific class was used or not using \@ifclassloaded{<class>}{<true>}{<false>} (already mentioned in a previous answer).

Also keep in mind that a class can be based on another class by loading it with \LoadClass or \LoadClassWithOptions. In this case \@ifclassloaded will be true for both. However AFAIK the standard classes are not based on any other classes.