... are so called because they were discovered by Galileo (in 1610). He was the first person to point a telescope at the skies, and the four biggest moons of Jupiter were the first objects to be discovered by telescope (unless you count things like craters on the Moon, which Galileo had examined in 1609.) They were also the first objects to be observed to orbit a body other than the Earth, and this dealt a significant dealt a blow to the then–accepted belief that the Earth was at the center of the universe and all other celestial bodies revolved around it.
They're named after various lovers of Zeus, in Greek mythology. I've listed them in order of orbital radius – or, if you prefer, distance from Jupiter (Io, Europa, Ganymede, Callisto). My trick for remembering the order is to think of two famous actors: Idris Elba and George Clooney. If you need to remember them in order of size, just think of the two actors the other way round: George Clooney and Idris Elba.
There is of course a potential issue here in that you need to remember which is the order of radius and which is the order of size. Well ... if you've got as far as learning either of these things, you probably know that Ganymede is the biggest moon in the Solar System.
Note that these are not the nearest moons to Jupiter. Last time I checked (April 2017) Jupiter had 67 known moons, and four of them had orbital radii smaller than that of Io. The largest of them is Amalthea - third one out - which is approximately 250 km across in its largest dimension (compared with Europa's 3,122 km and Ganymede's 5,262).
© Haydn Thompson 2017