This is an example of an Iris. It is popular in flower gardens. This particular species is Iris fernaldii, whomon name is Fernald's iris. The family Iridaceae is named for the The genus Iris, so this is a good place to start.
Crocus is another genus in the Iridaceae. It is also widely planted in flower gardens. This species, Crocus sativus is also very well known, as it is the source of the spice saffron, the most expensive spice in the world. It doesn't really have a common name. People just use "crocus."
Gladiolus is yet another popular garden flower, well known for it's huge tower of blooms. This species is Gladiolus 'Jubilee' Gladiolus x hortulanus. Instead of a common name, people just use "gladiolus."
Babiana rubrocyanea
Moraea vegeta