What is the process of attracting a mate called?

The most appropriate I could find was 'courtship'. Another possibility could be 'display'.

Wikipedia says: Courtship display is a special, sometimes ritualised, set of behaviours which some animals perform as part of courtship. Courtship behaviours can include special calls, postures, and movements, and may involve special plumage, bright colours or other ornamentation. A good example is the 'dancing' done by male birds of paradise. Display is a form of animal behaviour, linked to survival of the species in various ways. One example of display used by some species can be found in the form of courtship, with the male usually having a striking feature that is distinguished by colour, shape or size, used to attract a female.