If the velocity of a charge is parallel to the magnetic field the charge moves with constant velocity because there's no net force.

If the velocity of a charge is perpendicular to the magnetic field the path is circular because the force is always perpendicular to the velocity.

What happens when the velocity is not one of these special cases, but has a component parallel to the field and a component perpendicular to the field?

The parallel component is associated with straight-line motion. The perpendicular component is associated with circular motion. The net motion is a combination of these, so the net motion is a spiral (also known as a helix).