Koi actually means carp in Japanese, referring to both wild and ornamental forms. The ornamental forms have been selectively bred from common carp showing color mutations, similar to the modern goldfish (another member of the carp family) and the long-finned colorful betta.
Koi usually is used for these guys specifically in English, but carp refers to a whole family of species; technically that would include koi and goldfish though the common carp is the first to spring to mind. Either way works for me, I love the common carp too even if they don't come in such pretty colors.