an exomoon is a natural satellite which orbits around an exoplanet, a planet which orbits around another star besides our sun. no exomoons have yet been discovered because they are too small to detectable by current technology. however, there existence is very plausible, because many moons occur in our solar system. they can be similar to our moons, lifeless and cold, asteroid sized to ganymedian size, but also can be as large as Earth, can have a sustainable atmosphere, possible occurring in the habitable zone, and possibly habitable, even more common than a habitable planet like our own.
Exomoon.
There's no special name. Stars also come 'out' during the day but they are too dim to overcome the bright blue sky. With binoculars you can see Venus in the day if you know exactly where to look (and don't point the binoculars at the Sun!).