A quick overview of the story:

Balac, the King of Moab, was concerned about Israel's victories against the Amorrhites, so he sent messengers with gifts to the diviner Balaam, trying to induce Balaam to curse Israel (Balaam was known for his effective curses and blessings). The confused, untrustworthy, idolatrous Balaam -- a man whom St. Peter described as being one who "loved the wages of iniquity" (II Peter 2:15) -- prayed about whether to go or not, was told not to, and sent the messengers on their way.

Balac them sent more messengers, offering Balaam greater gifts in return for a cursing of Israel. Balaam prayed again, and this time was told to go -- provided that He did all that God told him to do -- but Balaam had decided sell out Israel at this point, and God knew it and was angry.

At any rate, the next morning, Balaam saddled up his ass and rode with the messengers to see Balac. En route, the ass grew distressed and swerved off the path. Balaam beat her.

Then the ass crushed Balaam's leg against a wall and Balaam beat her again.

Then the ass sank to the ground, refusing to budge. Balaam beat her once more -- and then "the Lord opened the mouth of the ass, and she said: 'What have I done to thee? Why strikest thou me, lo, now this third time?'"

Balaam answered, "Because thou hast deserved it, and hast served me ill: I would I had a sword that I might kill thee."

The ass said, "Am not I thy beast, on which thou hast been always accustomed to ride until this present day? tell me if I ever did the like thing to thee."

Balaam replied, "Never" -- and then suddenly his eyes were opened and he saw why the ass behaved as she did: an angel stood in the path, with sword drawn, barring passage. The angel upbraided Balaam for his cruelty to the ass and told him that the ass saved his life. Balaam was allowed to continue on, but only if he did what God commanded.
 

Epilogue:

Balaam met with Balac and, though he committed the sin of idolatry with the King, refused to curse Israel. Twice more Balac tried to get Balaam to curse Israel, and twice more Balaam refused. Instead, in spite of his evil, he blessed Israel and, before leaving for home, related a Messianic prophecy: "I shall see him, but not now: I shall behold him, but not near. A STAR SHALL RISE out of Jacob and a sceptre shall spring up from Israel: and shall strike the chiefs of Moab, and shall waste all the children of Seth. And he shall possess Idumea: the inheritance of Seir shall come to their enemies, but Israel shall do manfully. Out of Jacob shall he come that shall rule, and shall destroy the remains of the city."

Balaam later led Israel into idolatry by sending women to seduce the men of Israel away from the faith. God punished Israel for this by plague and war -- a war in which Balaam got his comeuppance and was slain (Numbers 31).

 
The Story as Told in Sacred Scripture
Numbers 22-24

And they went forward and encamped in the plains of Moab, over against where Jericho is situate beyond the Jordan. And Balac the son of Sephor, seeing all that Israel had done to the Amorrhite, And that the Moabites were in great fear of him, and were not able to sustain his assault, He said to the elders of Madian: So will this people destroy all that dwell in our borders, as the ox is wont to eat the grass to the very roots.

Now he was at that time king in Moab. He sent therefore messengers to Balaam the son of Beer, a soothsayer, who dwelt by the river of the land of the children of Ammon, to call him, and to say: Behold a people is come out of Egypt, that hath covered the face of the earth, sitting over against me. Come therefore, and curse this people, because it is mightier than I: if by any means I may beat them and drive them out of my land: for I know that he whom thou shalt bless is blessed, and he whom thou shalt curse is cursed.

And the ancients of Moab, and the elders of Madian, went with the price of divination in their hands. And when they were come to Balaam, and had told him all the words of Balac: He answered: Tarry here this night, and I will answer whatsoever the Lord shall say to me.

And while they stayed with Balaam, God came and said to him: What mean these men that are with thee? He answered: Balac the son of Sephor king of the Moabites hath sent to me, Saying: Behold a people that is come out of Egypt, hath covered the face of the land: come and curse them, if by any means I may fight with them and drive them away. And God said to Balaam: Thou shalt not go with them, nor shalt thou curse the people: because it is blessed.

And he rose in the morning and said to the princes: Go into your country, because the Lord hath forbid me to come with you.

The princes returning, said to Balac: Balaam would not come with us. Then he sent many more and more noble than he had sent before: Who, when they were come to Balaam, said: Thus saith Balac the son of Sephor, Delay not to come to me: For I am ready to honour thee, and will give thee whatsoever thou wilt: come and curse this people.

Balaam answered: If Balac would give me his house full of silver and gold, I cannot alter the word of the Lord my God, to speak either more or less. I pray you to stay here this night also, that I may know what the Lord will answer me once more.

God therefore came to Balaam in the night, and said to him: If these men be come to call thee, arise and go with them: yet so, that thou do what I shall command thee.

Balaam arose in the morning, and saddling his ass went with them. And God was angry. And an angel of the Lord stood in the way against Balaam, who sat on the ass, and had two servants with him. The ass seeing the angel standing in the way, with a drawn sword, turned herself out of the way, and went into the field. And when Balaam beat her, and had a mind to bring her again to the way, The angel stood in a narrow place between two walls, wherewith the vineyards were enclosed. And the ass seeing him, thrust herself close to the wall, and bruised the foot of the rider. But he beat her again: And nevertheless the angel going on to a narrow place, where there was no way to turn aside either to the right hand or to the left, stood to meet him. And when the ass saw the angel standing, she fell under the feet of the rider: who being angry beat her sides more vehemently with a staff. And the Lord opened the mouth of the ass, and she said: What have I done to thee? Why strikest thou me, lo, now this third time?

Balaam answered: Because thou hast deserved it, and hast served me ill: I would I had a sword that I might kill thee.

The ass said: Am not I thy beast, on which thou hast been always accustomed to ride until this present day? tell me if I ever did the like thing to thee.

But he said: Never.

Forthwith the Lord opened the eyes of Balaam, and he saw the angel standing in the way with a drawn sword, and he worshipped him falling flat on the ground. And the angel said to him: Why beatest thou thy ass these three times? I am come to withstand thee, because thy way is perverse, and contrary to me: And unless the ass had turned out of the way, giving place to me who stood against thee, I had slain thee, and she should have lived.

Balaam said: I have sinned, not knowing that thou didst stand against me: and now if it displease thee that I go, I will return. The angel said: Go with these men, and see thou speak no other thing than what I shall command thee. He went therefore with the princes.

And when Balac heard it he came forth to meet him in a town of the Moabites, that is situate in the uttermost borders of Arnon. And he said to Balaam: I sent messengers to call thee, why didst thou not come immediately to me? was it because I am not able to reward thy coming?

He answered him: Lo, here I am: shall I have power to speak any other thing but that which God shall put in my mouth?

So they went on together, and came into a city, that was in the uttermost borders of his kingdom. And when Balac had killed oxen and sheep, he sent presents to Balaam, and to the princes that were with him.

And when morning was come, he brought him to the high places of Baal, and he beheld the uttermost part of the people.

And Balaam said to Balac: Build me here seven altars, and prepare as many calves, and the same number of rams. And when he had done according to the word of Balaam, they laid together a calf and a ram upon every altar.

And Balaam said to Balac: Stand a while by thy burnt offering, until I go, to see if perhaps the Lord will meet me, and whatsoever he shall command, I will speak to thee.

And when he was gone with speed, God met him. And Balaam speaking to him, said: I have erected seven altars, and have laid on everyone a calf and a ram.

And the Lord put the word in his mouth, and said: Return to Balac, and thus shalt thou speak.

Returning he found Balac standing by his burnt offering, with all the princes of the Moabites: And taking up his parable, he said: Balac king of the Moabites hath brought me from Aram, from the mountains of the east: Come, said he, and curse Jacob: make haste and detest Israel. How shall I curse him, whom God hath not cursed? By what means should I detest him, whom the Lord detesteth not? I shall see him from the tops of the rocks, and shall consider him from the hills. This people shall dwell alone, and shall not be reckoned among the nations. Who can count the dust of Jacob, and know the number of the stock of Israel? Let my soul die the death of the just, and my last end be like to them.

And Balac said to Balaam: What is this that thou dost? I sent for thee to curse my enemies: and thou contrariwise blessest them.

He answered him: Call I speak any thing else but what the Lord commandeth?

Balac therefore said: Come with me to another place from whence thou mayest see part of Israel, and canst not see them all: curse them from thence. And when he had brought him to a high place, upon the top of mount Phasga, Balaam built seven altars, and laying on every one a calf and a ram, He said to Balac: Stand here by thy burnt offering while I go to meet him.

And when the Lord had met him, and had put the word in his mouth, he said: Return to Balac, and thus shalt thou say to him. Returning he found him standing by his burnt sacrifice, and the princes of the Moabites with him. And Balac said to him: What hath the Lord spoken? But he taking up his parable, said: Stand, O Balac, and give ear: hear, thou son of Sephor: God is not a man, that he should lie, nor as the son of man, that he should be changed. Hath he said then, and will he not do? hath he spoken, and will he not fulfill? I was brought to bless, the blessing I am not able to hinder.

There is no idol in Jacob, neither is there an image god to be seen in Israel. The Lord his God is with him, and the sound of the victory of the king in him. God hath brought him out of Egypt, whose strength is like to the rhinoceros. There is no soothsaying in Jacob, nor divination in Israel. In their times it shall be told to Jacob and to Israel what God hath wrought. Behold the people shall rise up as a lioness, and shall lift itself up as a lion: it shall not lie down till it devour the prey, and drink the blood of the slain.

And Balac said to Balaam: Neither curse, nor bless him.

And he said: Did I not tell thee, that whatsoever God should command me, that I would do?

And Balac said to him: Come and I will bring thee to another place; if peradventure it please God that thou mayest curse them from thence.

And when he had brought him upon the top of mount Phogor, which looketh towards the wilderness, Balaam said to him: Build me here seven altars, and prepare as many calves, and the same number of rams. Balac did as Balaam had said: and he laid on every altar, a calf and a ram.

And when Balaam saw that it pleased the Lord that he should bless Israel, he went not as he had gone before, to seek divination: but setting his face towards the desert, And lifting up his eyes, he saw Israel abiding in their tents by their tribes: and the spirit of God rushing upon him, He took up his parable and said: Balaam the son of Beor hath said: The man hath said, whose eye ire stopped up: The hearer of the words of God hath said, he that hath beheld the vision of the Almighty, he that falleth, and so his eyes are opened: How beautiful are thy tabernacles, O Jacob, and thy tents, O Israel! As woody valleys, as watered gardens near the rivers, as tabernacles which the Lord hath pitched, as cedars by the waterside. Water shall flow out of his bucket, and his seed shall be in many waters. For Agag his king shall be removed, and his kingdom shall be taken awry. God hath brought him out of Egypt, whose strength is like to the rhinoceros. They shall devour the nations that are his enemies, and break their bones, and pierce them with arrows. Lying down he hath slept as a lion, and as a lioness, whom none shall dare to rouse. He that blesseth thee, shall also himself be blessed: he that curseth thee shall be reckoned accursed.

And Balac being angry against Balaam, clapped his hands together and said: I called thee to curse my enemies, and thou on the contrary hast blessed them three times. Return to thy place. I had determined indeed greatly to honour thee, but the Lord hath deprived thee of the honour designed for thee.

Balaam made answer to Balac: Did I not say to thy messengers, whom thou sentest to me: If Balac would give me his house full of silver and gold, I cannot go beyond the word of the Lord my God, to utter any thing of my own head either good or evil: but whatsoever the Lord shall say, that I will speak? But yet going to my people, I will give thee counsel, what this people shall do to thy people in the latter days. Therefore taking up his parable, again he said: Balaam the son of Beor hath said: The man whose eye is stopped up, hath said: The hearer of the words of God hath said, who knoweth the doctrine of the Highest, and seeth the visions of the Almighty, who falling hath his eyes opened: I shall see him, but not now: I shall behold him, but not near. A STAR SHALL RISE out of Jacob and a sceptre shall spring up from Israel: and shall strike the chiefs of Moab, and shall waste all the children of Seth. And he shall possess Idumea: the inheritance of Seir shall come to their enemies, but Israel shall do manfully. Out of Jacob shall he come that shall rule, and shall destroy the remains of the city. And when he saw Amalec, he took up his parable, and said: Amalec the beginning of nations, whose latter ends shall be destroyed.

He saw also the Cinite: and took up his parable, and said: Thy habitation indeed is strong: but though thou build thy nest in a rock, And thou be chosen of the stock of Cin, how long shalt thou be able to continue? For Assur shall take thee captive.

And taking up his parable, again he said: Alas, who shall live when God shall do these things? They shall come in galleys from Italy, they shall overcome the Assyrians, and shall waste the Hebrews, and at the last they themselves also shall perish. And Balaam rose, and returned to his place: Balac also returned the way that he came.