A simple system to describe how 'far' you have gone with a girl. It refers to bases as markers of your sexual exploits. From first base to fourth, this alliterary device explains the 'distances' of each base: Frenchie, feel, finger, f***.
There are many phrases that go with it, such as 'Pitch Runner', which is describing when you skip a base, and Homer, which is going straight to fourth base, or sex. Run out means being stopped before getting to a base.
Obviously, there is some discrepancy, like blowjobs and handjobs obviously coming in somewhere between third and fourth.
Person 1: How far'd you get with Jenny last night?
Person 2: Err...well, we-
Person 1: Baseball System, man.
Person 2: (quietly) ran out before first.