Anton, I believe that machine uses steppers. With the machine powered off (and if possible the motor disconnected) can you slide the head along the axis smoothly? If not, check the gears and belts for damage.
If so, the problem is most likely wiring or the motor driver (it could be an open coil in the motor but that is unlikely). The wiring should be easy to check, and if that's ok the motor driver IC (L297/298) or clamp diodes have most likely failed.
You can switch the X and Y axis cables to test it, if the problem moves to the X axis the driver is almost certainly the problem.