What is a pink flower called?

That their flowers typically are pink in color and have frilled (pinked) margins is the reason why the flowers are called "pinks".

Specifically, the flower in question belongs to the genus Dianthus. It has worldwide appeal even though it is predominantly native to Asia and Europe. It is immediately recognizable by beautifully pale to dark pinks colors (although there are other colors such as the yellow of D. knappii) and interesting margins which look as through they were trimmed by pinking shears.