Wall socket - plug - fuse - cable (especially where it bends coming out of plug) on/off switch - thermal cut out - motor.
Most of the vacuum cleaner faults I have looked at have been a broken conductor inside the cable near the plug. A quick and cheap repair to chop off 6" or so and fit a new plug - and also an easy elimination of things more expensive to repair.
howtomendit.com for free repair help, information and advice.
Try fixfault.com to fix vacuum cleaners.