The scientific name for this fossilized rock is hexagonaria percarinata. In Michigan, this gray rock is known as a Petoskey stone, named for the city of Petoskey, Michigan, where many of the stones were found. Petoskey was named for the Ottawa Indian chief Petosegay or Pet-O-Sega, which means "Rays of the Rising Sun." While Chief Pet-O-Sega was named before the stone was associated with him, some see its significance, as if the fossilized coral polyps are suns and the fossilized tentacles are their rays. In 1965, Michigan declared its state stone the Petoskey Stone, becoming the first state to set a fossil as its state stone or rock. Ella Jane Petoskey, the only living grandchild of Chief Pet-O-Sega, was present as the governor signed the State Stone bill.
Petoskey stones are softer than typical agates because they are fossils, a delicate 3-3.5 on Mohs Scale, but when you find them on the beach, you will discover that they are very smooth. This is because they have been naturally tumbled by rushing water, sand, and other stones. Petoskeys can be hand-polished with sandpaper and corduroy (explore p. 4 of this PDF download for a step-by-step instruction, by the Michigan Department of Environmental Quality, Geological Survey Division)
Where else can you find fossil agate coral? In the US, Florida's state stone, Agatized Coral, which it chose in 1979, is a form of silicified coral more similar to petrified wood. This is the only gemstone naturally found in Florida, and can be located near Tampa Bay.