Isaac Newton (1643-1727) proved mathematically that Galileo was correct. Newton's law of universal gravitation states that the attraction between two objects M and m placed a distance r apart (from center to center) is given by F = GMm/r2 , where G is the gravitational constant. But Newton's second law states that the force (F) on an object (m) is related to it's acceleration (a) by F = ma. Combining these two equations we get F = GMm/r2 = ma, or a = GM/r2. Thus, the gravitational acceleration of an object is only dependent on the mass of the Earth (M), it's distance from the Earth's center (r), and the gravitational constant G. We call this acceleration due to gravity g to distiguish it from acceleration due to some other force. The precise value of g varies with location and the local Earth density, but standard gravity is defined as g = 9.80665 m/s2.
Newtons third law states that forces exist in equal and opposite pairs. This means that for a given force F1 acting on a small mass m resulting in an acceleration a1, the Earth also experiences a force F2 on its mass M resulting in an acceleration a2. The total acceleration of the Earth and the small mass m is just the sum a1 + a2 = G(M + m)/r2. However, the mass of the Earth (5.98 x 1024 kg) is many many orders of magnitude greater than the small mass m so M + m is essentially M and a2 is essentially 0. If the mathematics are hard to read here, check out the Wikipedia articles on Universal Gravitation and Earth's gravity.
This experiment has even been performed on the moon by Apollo 15 Commander David Scott using a 1.32 kg geological hammer and a 0.03 kg falcon feather (Falcon was the name of the lunar module). Because there's no atmosphere on the moon, the hammer and the feather landed on the lunar surface simultaneously. This was broadcast live in 1971 and a link to the video is below.
On a historical note, the guinea was the British gold coin first used in this experiment and in the US it's sometimes called the Penny and Feather. Many current versions use a piece of ordinary metal instead of a coin, and a piece of paper instead of a feather.
Please note that the title picture is from the PIRA 200 demonstration list and is copyright 1993 by the University of Minnesota and the Physics Instructional Resource Association. According to this webpage, the title image is released for nonprofit educational use only. All pictures used for other nonprofit purposes, need to retain the signature of their creators...