Above all, output, adjustability weight and size. Then one may consider shape and color as well.
For medium or longer hair, only consider stronger hairdryers (at least 1800W).
Two versions are distinguished: one with AC and one with DC motor. It is easy to tell them apart, just feel their weight. AC motors are manufactured without carbon brush and usually have a longer lifetime with higher output, though they are notably heavier. Hairdryers with DC motors are not suitable for industrial use but are more comfortable to use.
There are hairdryers equipped with ionization technology. This technology negates the electrostatic charge thus turning the hair silkier and healthier.
In case of shorter hair or for travelling a smaller hairdryer should be enough.