Before iOS5, Mobile Me was the way Find My iPhone operated. It was much the same as with iCloud. When iOS5 was introduced, those on Mobile Me (which was discontinued) didn't need to do anything - the Find My iPhone ability was switched over automatically.
Originally find my phone only worked with a fully paid MobileMe subscription. Apple then allowed free MobileMe accounts for anyone but these were limited to find my phone functionality only (no email or syncing contacts etc). Same app works for iCloud or MobileMe accounts