Galileo first observed the moons of Jupiter in 1610. Using a telescope at Padua, he noticed tiny star-like points near Jupiter that changed position from night to night.
His first recorded observation came on January 7, 1610, when he saw three apparent stars arranged near Jupiter. His instrument was not powerful enough to separate Io and Europa, whose combined light appeared as one point. On the following night, January 8, he saw them as distinct objects. By January 13, he had viewed all four moons together, and continued observations convinced him that they were orbiting Jupiter rather than remaining fixed background stars.
Galileo published the discovery in Sidereus Nuncius in March 1610. The finding was revolutionary because it demonstrated that not everything in the heavens revolved around Earth, challenging the traditional geocentric model. The four bodies are now called Io, Europa, Ganymede, and Callisto, collectively the Galilean moons.
Some accounts mention late December 1609 because dates in the period can be converted between the Julian and Gregorian calendars. Galileo’s own published and commonly used date is January 1610, making 1610 the fair canonical answer.