The golden calf (עגל הזהב) was an idol (a cult image) made for the Israelites during Moses' absence, as he went up to Mount Sinai. According to the Hebrew Bible, the calf was made by Aaron to satisfy the Israelites, whereas the Quran indicates the maker to be Samiri.
In Hebrew, the incident is known as "Chet ha'Egel" (חטא העגל) or "The Sin of the Calf". It is first mentioned in Exodus 32:4 (Taha 20:83 in the Quran). In Egypt, whence the Hebrews had recently come, the Apis Bull was the comparable object of worship, which the Hebrews were reviving in the wilderness. Among the Egyptians' and Hebrews' neighbors in the Ancient Near East and in the Aegean, the Aurochs, the wild bull, was widely worshipped, often as the Lunar Bull and as the creature of El. Its Minoan manifestation survived as the Cretan Bull of Greek myth.
When Moses went up onto Mount Sinai to receive the Ten Commandments (Exodus 19:20), he left the Israelites for forty days and forty nights (Exodus 24:18). The Israelites feared that he would not return and asked Aaron to make gods for them (Exodus 32:1). The Bible does not note Aaron's opinion of this request, merely that he complied, and gathered up the Israelites' golden earrings. He melted them and constructed the golden calf. The Hebrew word calf can also mean circle. This might seem to indicate that they made a RA symbol rather than cast a calf, which would require a difficult manufacturing process with furnace, beeswax, clay, etc.
Aaron also built an altar before the calf / circle (RA), and the next day, the Israelites made offerings and celebrated.
The Lord told Moses that his people had corrupted themselves, and that he planned to eliminate them, but Moses argued and pleaded that they should be spared (Exodus 32:11); the Lord relented. Moses went down from the mountain, but upon seeing the calf, he too became angry. He threw down the tablets upon which God's law had been written, and broke them. Moses then burnt the golden calf in the fire, ground it to powder, scattered it on water, and forced the Israelites to drink it. He questioned Aaron about the event, who admitted to collecting the gold, throwing it into the fire, and out came a calf. Then Moses gathered the sons of Levi and set them to slaying a large number of adult males (3000). A plague then struck the Israelites. Nevertheless, the Lord stated that he would one day visit the Israelites' sin upon them.
Since Moses had broken the tablets, the Lord instructed him to return to Mount Sinai yet again (Exodus 34:2) to receive a replacement.
Many Christian scholars have suggested that the Israelites were worshipping the Egyptian god Apis, falling back into what they had known for centuries while in captivity. It is suggested that the "idolatry" (a voiding of the Second Commandment) on display here was the worship of another god. However, forging an image of Apis would not have violated the Second Commandment before it had literally violated the First, "worship no other gods".
As such, what may have actually transpired within the event is that the Israelites had not so much voided the First Commandment so much as they had violated the Second, which prohibited the making of an image of Yahweh. In Exodus 32 it states: "When Aaron saw this, he built an altar in front of the calf and announced, 'Tomorrow there will be a festival to the LORD (Yahweh).'" Within the context of the Exodus story, it would be highly unlikely that the Israelites, after witnessing the miracles of the Exodus first hand, would have fallen into the worship of another god immediately after Yahweh had just spoken the Decalogue in their midst.
Complicating the matter still further, there are indications in the story that the Israelites intended a replacement Moses rather than a replacement or even icon for God Himself. For instance, the rationale they gave Aaron for making the thing in the first place (Ex. 32:1) was "because this the man Moses, who took us out of the land of Egypt, we do not know what has become of him."
According to Exodus 32:4 the golden calf is made "...And they exclaimed, "This is your god O Israel, who brought you up from the land of Egypt!"
The preceding paragraph contains an inaccurate quotation of Exodus 32:4. The verse reads 'these' just prior to 'Elohim-of-you,' which explicitly pluralizes the reference to Elohim, which is often used in singular or plural contexts in the bible (see the wiki article for Elohim).
However, later on in 1 Kings 12:28, Jeroboam tries to stop the Northern Israelites from visiting Jerusalem. He has two high places erected at Dan and Bethel as new offering places. At each of these he has constructed a golden calf and says "It is too much for you to go up to Jerusalem. Here are your gods, O Israel, who brought you up out of Egypt." A similar phrase.
The creation of the golden calves may have been an attempt to identify the Lord with Baal. Among the Phoenicians, Baal was sometimes called the "calf" whereas the supreme god El (God) was called the "bull". Bovine whole-burnt offerings were an important part of Baal worship. The golden calf may have been a zoomorphic ark for Baal, just as winged lions (cherubim) were for the Lord. By making a calf pedestal, instead of a lion, would have been an attempt to identify the Lord with the Canaanite son of Dagon, Baal.
The construction of the two golden calves would have been seen as a gross blasphemy by the Kings author, on a par with the original Golden Calf episode. The Levite priests in the North would have found those golden calves an irritation as they were looked after by non-Levite priests, and were probably seen as idolatrous. A reference to the original golden calf episode may have been seen fitting. There may even have been some cross over of the language.
The Quranic Version of the episode is similar in most respects, except that the golden calf is constructed by a man named Samiri, rather than Aaron. Samiri claims that Moses has disappeared, and the Israelites have to find a new god. To this end, Samiri makes a golden calf from the gold jewelry brought out of Egypt.
Aaron, who is acting as leader in Moses' absence, attempts to prevent them from worshipping the statue, but is unsuccessful. When Moses does return, he is initially infuriated at the pagan ritual and Aaron's inability to stop it. Moses then exiles Samiri and orders the golden calf burnt and its ashes cast into the sea.