The purpose of the engine relay is to keep accessories that are designed to operate with the engine running, only operate with the engine running. It's used to reduce engine-off load on the battery, as well as prevent the pump from running all of the time (at least with me, I leave it in ON so I can listen to the radio and not here the damn beeping; I'm sure others do it too) and from burning up.
You'll notice the blower motor will run as well as the fuel pump. However, the lights on the dash will not fully go out. This is a diagnostic/last resort/emergency measure. If the alternator is not charging, you'll run your battery down.