Be sure to specify the folder "C:\ProgramData\ENIGMA"?
Technically it can be anywhere, but without any spaces. That is why it's not "Program files" or something like that. ProgramData is a place in >WinXP just for this. In WinXP it tries to go to places with spaces and that breaks the compiler.
What it holds is all of the compiled files for ENIGMA. The first time you run ENIGMA it will compile not only your game, but also the ENIGMA game engine itself (+ all the selected systems like sound and graphics and selected extensions). They are put in .eobj folder in "C:\ProgramData\ENIGMA" (or anywhere you specify). The reason it puts it there and not in TEMP for example, is because technically it has to do it only once. So no reason for the files to be deleted and then compiled again (which can take time on older hardware). Then there is also Preprocessor_Environment_Editable folder inside it, which holds your games code translated into C++. It doesn't hold your resources (like sprites or sound files), but only meta information about them (like settings in sprites - for example - the origin position or the number of frames). This information is then compiled (and put in .eobj folder as well) and used together with the ENIGMA engine to create your final exe. It also holds some debug information (like compile logs).
So when you load a new project the content in Preprocessor_Environment_Editable will change, but most of the stuff in .eobj will stay the same. Your game (the code you write) is basically compiled into the SHELLmain.o file inside the .eobj folder.