If you want to copy a shapefile (or file geodatabase for that matter) without exiting all Arcgis programs or killing processes open up a Command Prompt window and:
xcopy /c some-shapefile.* d:\some\other\place
The /c means continue even if there are errors. More info about parameters on SS64.
Use the Command Prompt here windows explorer extension (vista and win7 users can just press shift when r-clicking) so you can open a prompt quickly in the folder you need to work in.