I've photographed this flower a few days ago in New York City:
Does anybody know the name of it? (and how is the orange part called?)
It's Echinacea. I've linked to one site but if you run an images search with Echinacea as the search term you'll see lots of examples.
Echinacea are members of the Compositae. The flower (the head) in your picture is actually made up of lots of individual smaller flowers (i.e. it is a composite flower). The petals are formed by ray flowers, and the central part, the disc is made up of disc flowers. More details at the WP page I have linked to for Compositae.