So Moses was a prophet and Aaron was a priest, and somehow, this led the Israelites to ask Moses to speak to the Lord for them...and to ask Aaron to make them a god.
To answer that question, we first have to understand the inherent difference between a prophet and a priest. Both speak the word of God. Both are called to mediate, in a way, between people and God (or God and people). Both are generally members of the community which they are called to serve. Both do amazing, incredible things as signs of God's favor being with them.
But a prophet is given for the people, while a priest is given to them.
The prophet speaks God's words for the benefit of the people. His job is to warn the people, to tell them the truth they need to hear, whether they want to hear it or not. He doesn't mince words. What the prophet does for the people is very raw. It's very coarse. It's no-holds-barred truth telling, a warning about the perils that lay before them if they do not change their way. A prophet's job is to stir the hearts of God's people.
A priest's job, however, is to sooth the hearts of God's people. His is a ceremonial office; it requires more decorum and tact than the prophet's. The priest mediates the rituals between God and the people more than the truth between them. He offers sacrifices, makes atonement, burns incense. He consecrates and blesses the people. In other words, he does everything that helps the people to feel more connected to God. As such, his role is less raw and rough; it's more tender.
At this point, you might be saying that perhaps I have it backward - that clearly, Aaron was the prophet (bringing a new god to the people) and Moses was the priest (interceding with the Lord on their behalf). Hang with me; I'm not done.
People don't, in general, like a prophet. Herod said about John the Baptist that the king was disturbed every time the prophet spoke to him. That's the feeling that people get with prophets. Their words are often hard to swallow. Primarily because they are usually convicting. And the people also have the sense that the prophet's first loyalty is to God, not to them (and it is), so it's easy for people to feel a bit of a disconnect between themselves and a prophet.
Such it is. Again, because the prophet has been given for them, not given to them. (You know, like when your parents said "This is for your own good" and it was never really something you enjoyed or necessarily appreciated at the time. But you trusted them when they said this, and maybe as you got older, you realized they were right.)
People have much more fondness for a priest, however, and for this same reason: he is given to them. They feel like the priest is theirs. They know that his loyalty is to God, but they feel just as much loyalty from him toward them, if not more. Because of the tender, personal, intimate nature of the priest's work, there's this weird sort of affection that develops between the people and the priest. And it's not long before the lines become blurred.
It's not long before the people think that the priest who God has given to them is the one man of God who is for them.
The priest can make atonement for the people, so he must know something about atonement. The priest can offer sacrifices for the people, so he must know what is pleasing in a sacrifice. The priest can bless the people, so he must know how to shower good things on the people. The priest must then know best what the people want from God...and so it's only natural that it would be the priest who could form the god that the people want.
That's where we find Aaron. He's been serving Israel as a priest - mediating between them and God. Serving them, blessing them, atoning for them, offering their sacrifices. If anyone knows what Israel needs from God, it's Aaron. So when their Lord goes M.I.A. with the prophet on the mountain (the prophet who, by the way, they're not entirely happy with - more on that some other time), they turn to Aaron to fill their void. And their request is not "mediate between us and the Lord." No. Moses is probably already doing that.
Their request is: you know what we need from our god. Make it happen.
Enter the golden calf.