Caption: Galileo's (1564--1642) drawings of Jupiter and the Galilean moons (obtained from observations with early telescopes) published in the Sidereus Nuncius (1610, in English The Star Messenger) which is in Latin as you can see---"Die quarta hora ſecunda circa Iouem quator ſtabant Stellae ..." where ſ is NOT f, but long s = ſ. See Sidereus Nuncius (1610) online for the whole darn book.

The Galilean moons oscillated about Jupiter on approximately a single line. Obviously, they were natural satellites orbiting Jupiter and, from a Copernican perspective in which the Earth is a planet, they were moons of Jupiter.

The page from the Sidereus Nuncius shows the Galilean moons at different periods, and thus reveals their oscillation around Jupiter.

Galileo named the new moons the Medician Stars for his patrons the Medici---but posterity disposed of that homage and called them after Galileo himself.

The discovery of the Galilean moons was strong evidence for heliocentrism:

It showed that there were astro-bodies unknown to the ancient Greek astronomers and anyone else before the telescope. Therefore, clearly the Ancients did NOT know everything.
It showed that NOT all astro-bodies orbited Earth which disproved a key point of Aristotelian cosmology and the Ptolemaic system.
Actually, Galileo's discovery of the full phases of Venus proved that Venus orbited the Sun and very strongly suggested all the planets orbited the Sun---to those NOT committed to Aristotelian cosmology and/or the Ptolemaic system. However, this result was NOT in the Sidereus Nuncius (1610) and was only published in 1613 (see Wikipedia: Phases of Venus: History).
It showed that planets could have moons, and, therefore, the Earth could have its Moon and still be a planet. The idea that the Earth was planet was a shocking aspect of heliocentrism.
It showed a clear case of smaller astro-bodies orbiting a larger one. This is was consistent with the still-unknown physics of heliocentrism in which the planets (including the Earth) orbited the Sun which seemed to be by far the largest astro-body.
It showed a clear case where the larger the orbital radius the longer the orbital period. Again, this is was consistent with the still-unknown physics of heliocentrism in which the planet orbital periods increase with orbital radius. See the animation in the figure below illustrating the orbits of the 3 innermost Galilean moons.

WRONG

Modern science was just aborning in 1610, and so the strength of the evidence for heliocentrism from the Galilean moons was NOT completely obvious, maybe NOT even completely obvious to Galileo and Johannes Kepler (1571--1630).