If you want to copy a shapefile (or file geodatabase for that matter) without exiting all Arcgis programs or killing processes, stop any active Edit sessions or geoprocessing tasks, open up a Command Prompt window and:
Shapefile:
xcopy /c /v d:\projects\aaa\some-shapefile.* d:\some\other\place
The /c means continue even if there are errors (e.g. skip in-use lock files); /v = verify.
File geodatabase:
xcopy /c /i /v d:\projects\aaa\some-file.gdb\* d:\some\other\place\some-file.gdb
/i means assume destination is a folder, not a file.
More info about parameters on SS64.
Vista, Win7 and newer users press shift when r-clicking in folder to open a prompt quickly in that spot. XP users install the Command Prompt here windows explorer extension for same.
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update Aug 2015: Added exit edit session, file-gdb example.