The events leading up to this incident are very important in understanding the meaning of the Golden Calf and why it was such a grave sin on the part of the Israelite people.
According to the story, the Israelites, after having spent 400 years as slaves in Egypt, called to God for freedom. God sent Moses to be the one to facilitate their removal from the tyrannical reign of the Pharoahs.
At first, life in Egypt was good for the Israelites as they found it to be a place of refuge from the famine that had afflicted them in their home land. The Egyptians were hospitable hosts who allowed the Israelite people to live according to their own laws and customs. As the Egyptian Empire grew, as did the oppression which was especially directed at these "foreigners" who did not respect the laws and edicts of the Egyptian kings.
Moses was directed to appeal to the humanity of the Pharaoh in asking for the release of the Hebrew slaves. Pharaoh refused repeatedly, so God brought ten plagues on Egypt. Still Pharaoh would not relent. Only after the 10th and final plague - the death of all the Firstborn sons, did Pharaoh agree to release the slaves.
After the Hebrews sacrificed lambs, placing the blood on the doorposts of their homes to protect them from the 10th plague, they fled in the middle of the night, crossing the Red Sea and entering into the land of Sinai. It is important to note that the story says that they were all led directly by God himself who took the form of a pillar of fire by night and cloud/smoke by day.
God led them directly to the mountain at Horeb (Mt. Sinai) where they all heard a voice commanding them to ascend the mountain. As a whole the people refused because of their fear so they elected to send Moses as a representative who will receive the words of God and communicate them to the people.
This was the first of several mistakes that these newly-liberated people made in their path out of slavery and into freedom. The mountain was open to all who were willing and receive the "law" directly from the Creator who dwells within them. The mountain is the symbol of the conscience. But rather than be ruled from within, the Israelites had chosen a human ruler.
The people began to panic for Moses was gone for 40 days. They assumed that he had abandoned them. So they decided to build a Golden Calf idol so that God would speak to them through it - once again choosing an intermediary between man and the one true lawmaker.