Galileo didn't invent the telescope but he was the first to use the telescope to study the heavens systematically. What he had observed in the heavens rocked the very foundations of Aristotle's universe and the theological-philosophical worldview that it supported. It is said that what Galileo saw was so disturbing for some officials of the Church that they refused to even look through his telescope; they reasoned that the Devil was capable of making anything appear in the telescope, so it was best not to look through it. Galileo's observations with his newly fashioned telescope (which he fashioned a vastly superior model of one he had simply heard about from Holland) convinced him of the truth of Copernicus' sun-centered theory.
Galileo's astrology studies allowed him to observe the Sun through his telescope. Here, he saw that the Sun had dark patches, which we now call sun spots. He also observed motion of the sun spots indicating that the Sun was rotating on an axis. These "blemishes" on the Sun were contrary to the doctrine of an unchanging perfect substance in the heavens, and the rotation of the Sun made it less strange that the Earth might rotate on an axis, too, just lie the Copernican model. Galileo also observed four points of light that changed their positions with time around the planet Jupiter. He concluded that these were object in orbit around Jupiter. He later discovered these were the four brightest moons around Jupiter (which we now call the Galilean moons).
These observations showed that there were new things in the heavens that Aristotle and Ptolemy had not known about. They also demonstrated that a planet could have moons circling it that would not be left behind as the planet moved around its orbit.
Galileo also used his telescope to show that Venus went through a complete set of phases, just like the Moon. This was among the most important in human history because it provided the first conclusive observational proof that was consistent with the Copernican system but not the Ptolemaic system.
As noted above, Galileo's astrology studies also brought us other observations, such as showing that the planet were disks, not point of light, showing that the great "cloud" called the Milky Way was composed of enormous numbers of stars that had not been seen before, that the moon was not smooth, as once thought, but was covered by mountains and craters, and showing that the planet Saturn had "ears". This may seem strange, but we now know that Galileo's astrology studies discovered the rings of Saturn, his telescope was just not good enough to show them as more than extensions on either side of the planet!
Beginning with the Sidereus Nuncius in 1610, Galileo brought the Copernican issue before a wide audience. In his letters on sun spots and in his letter to the Grand Duchess Christina he actually interpreted the problematical biblical passage in the book of Joshua to conform to a heliocentric cosmology. Galileo's life was interesting, even for a man in his time. Galileo's astrology work related strongly to physics and was a pivotal figure in the development of modern astronomy. He provided the crucial observations that proved the Copernican hypothesis and also laid the foundations for a correct understanding of how objects moved on the surface of the earth and of gravity. This work, however, challenged the authority of the Church through his assault on the Aristotelian concept of the Universe and eventually got him into trouble with the Inquisition. Late in life, he was forced to recant his Copernican views and spend his last years under house arrest. Galileo's contributions including the "Law of Inertia" led Issac Newton, who was born in 1642, the year Galileo died laid the foundation for Newton's First Law of Motion.