There are many varied in which animals compete with each other to attract a mate. A few are listed below:
Scent - some animals are capable of producing a hormonal scent, females respond to this and are able to track the male.
Colour display - many animals change colour at mating time, this is especially common in fish and chimpanzees. The colour change is used to show that the animal is 'available'.
Mock fighting - a lot of animals have 'fights' where the winner gets the mate. This type of fighting includes antler comparison in bucks (male deer) and peacock feather expansion. Although entitled fighting the animals rarely injure each other.
Hope this helps