A magnetic field exerts a force on a single moving charge, so it's not surprising that it exerts a force on a current-carrying wire, seeing as a current is a bunch of moving charges.

F = qv ´ B

Using q = I * t this becomes:

F = Ivt ´ B

But a velocity multiplied by a time is a length L, so this can be written:

F = IL ´ B

The magnitude of the force is F = ILB sin(q).

The direction of the force is given by the right-hand rule, where your fingers point in the direction of the current. Current is defined to be the direction of flow of positive charges, so you don't have to worry about your right hand lying to you...it will always give the correct direction.