Why did God tell Abraham to sacrifice Isaac?

Genesis 22:1-12
Some time later God tested Abraham. He said to him, "Abraham!"
"Here I am," he replied.
Then God said, "Take your son, your only son, Isaac, whom you love, and go to the region of Moriah. Sacrifice him there as a burnt offering on one of the mountains I will tell you about."

Early the next morning Abraham got up and saddled his donkey. He took with him two of his servants and his son Isaac. When he had cut enough wood for the burnt offering, he set out for the place God had told him about. On the third day Abraham looked up and saw the place in the distance. He said to his servants, "Stay here with the donkey while I and the boy go over there. We will worship and then we will come back to you."
Abraham took the wood for the burnt offering and placed it on his son Isaac, and he himself carried the fire and the knife. As the two of them went on together, Isaac spoke up and said to his father Abraham, "Father?"
"Yes, my son?" Abraham replied.
"The fire and wood are here," Isaac said, "but where is the lamb for the burnt offering?"
Abraham answered, "God himself will provide the lamb for the burnt offering, my son." And the two of them went on together.
When they reached the place God had told him about, Abraham built an altar there and arranged the wood on it. He bound his son Isaac and laid him on the altar, on top of the wood. Then he reached out his hand and took the knife to slay his son. But the angel of the LORD called out to him from heaven, "Abraham! Abraham!"
"Here I am," he replied.
"Do not lay a hand on the boy," he said. "Do not do anything to him. Now I know that you fear God, because you have not withheld from me your son, your only son."

Deuteronomy 12:31 (see also Leviticus 20:2-5, Jeremiah 32:35, Ezekiel 20:26, Isaiah 57:4-5)
You must not worship the Lord your God in their way, because in worshiping their gods, they do all kinds of detestable things the Lord hates. They even burn their sons and daughters in the fire as sacrifices to their gods.

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Why would God command Abraham to sacrifice his own child?
First, we should realize what God was not doing:
God was not tempting Abraham. God was not enticing Abraham to do wrong, but was testing him to see if he would do what was right.
God was not instituting or condoning child sacrifice. As seen in Deuteronomy 12:31 and the other passages above, God abhors child sacrifice. It's important to remember that God prevented the sacrifice from actually occurring. He did not desire the sacrifice as an act of worship or for any other reason beyond testing Abraham.
God was not telling Abraham to do wrong. God has the right to take human life (see the article on God's moral authority) and could therefore authorize Abraham to do so in a particular case. Note that had Abraham decided of his own accord to sacrifice Isaac, he would have been wrong and his act would have been condemned by God (as were other human-initiated sacrifices).
Why then would God give this command? The point was for Abraham to demonstrate that he trusted God completely and placed him above all else, even his own son.¹ Though God of course already knew that Abraham had faith in him, it was necessary for Abraham to prove it through action. "His faith was made complete by what he did" (James 2:21-23). Because of his actions, not only God but Abraham, his family and future generations knew that Abraham trusted God. This trust was important because it indicated that Abraham had the proper relationship with God (he was treating God as God deserves to be treated) and could benefit from God's good plans for his life.
Wasn't it wrong for Abraham to obey God?
God's command to Abraham was not wrong, for God has the right to take human life (see the article on God's moral authority) and therefore had the right to command Isaac's death. Abraham had known God for many years prior to this, so he knew God's character and knew that God had the right to give this command. Had Abraham initiated the sacrifice or followed the order of someone who was not authorized by God, then his act would have been wrong.
Furthermore, Abraham knew that God had promised him offspring through Isaac, so this gave him reason to believe that God did not intend for Isaac to die permanently:
By faith Abraham, when God tested him, offered Isaac as a sacrifice. He who had received the promises was about to sacrifice his one and only son, even though God had said to him, "It is through Isaac that your offspring will be reckoned." Abraham reasoned that God could raise the dead, and figuratively speaking, he did receive Isaac back from death. (Hebrews 11:17-19)
Some have questioned whether Abraham truly loved Isaac if he was willing to sacrifice him. But the passage itself as well as other places in Genesis point out that Abraham did love Isaac:

• Isaac was the only child of his beloved wife Sarah, the son promised to him by God (Gen 17:19)
• He held a feast on the day Isaac was weaned (Gen 21:8)
• God refers to Isaac as "your only son, Isaac, whom you love" (Gen 22:2)
• He made a special effort to get Isaac a godly wife (Gen 24:1-4)
• Isaac received all of the inheritance (Gen 25:5)

While Genesis doesn't record Abraham's emotional reaction to God's command, that does not mean carrying it out was an easy thing for him to do.