Let me start from your comment on Lubos' answer:
If we have a electron near a coil of wire that has current running through it, certainly the electron will move a certain direction right?
No, it's not that simple. For a given coil of wire producing a given magnetic field, the electron can experience a force in any direction that is perpendicular to the field. It depends on which way the electron is moving. (The force is always perpendicular to both the field and the electron's velocity.) In fact, if the electron is just sitting at rest, or is moving parallel to the magnetic field, it experiences no force at all.
You might be confused because you're thinking of the electrostatic force. That one is always parallel to the electric field; it doesn't matter how the particle is moving, and that's why you can draw electrostatic force lines. But that doesn't work with the magnetic force.