Do I have to leave an application running in order for it work? For example: I installed a pedometer application do I need to leave it on for it to count my steps?
That depends on the app, to be honest. Try using the Pedometer, press home, walk around, go back to the pedometer and see if the number changed.
If not, then you need to let it fall asleep with the app still there, if so, then you can go about doing other things with your phone and the pedometer.