A telescope. In 1610, Italian scientist and inventor Galileo put a couple of curved pieces of glass together in a tube, and discovered he could see farther than the human eye allowed. When he observed Jupiter, he was delighted to discover that it had 4 moons, which are still called Jupiter's "Galilean" moons even today. They are, in order of increasing distance from Jupiter: Io, Europa, Ganymede, and Callisto.
Yes, it is possible to see Jupiter's four largest moons, known as the Galilean moons, with just a pair of binoculars or even with the naked eye. These moons are Io, Europa, Ganymede, and Callisto.
You can see Jupiter's moons any night whenever you can see Jupiter, with the possible exception of times when the Moon is close to it. Just now (2014) Jupiter is mostly visible in midwinter.
With a telescope
Please see related question for a complete listing of Jupiter's 40+ moons.
You would not be able to see the moons from the surface; Jupiter's atmosphere is too thick.
The moon we see doesn't, but all of Jupiter's moons do. (About 50)
Jupiter has 63 confirmed moons. See link for further information about them.
Galileo was able to see only Jupiter's largest moons, Io, Europa, Ganymede, and Callisto, because these moons are the largest and brightest of Jupiter's many satellites. Their size and brightness made them visible using the relatively primitive telescope technology available to Galileo in the 17th century. Smaller moons may have been too faint to be seen with his telescopes.
He saw 4 moons
When you use a telescope or binoculars. Jupiter has moons which are often easily visible in binoculars.
Jupiter's many moons
Our moon is the only moon visible from earth with the naked eye. With a good personal telescope, it is possible to see 4 of Jupiter's moons.