I want to know the story of Lord Shiva and demon Andhakasura. What is his story?
The story of Andhakasura is told in Vishnu Parva of Harivamsha of Mahabharata in chapter 86 and chapter 87.
Here is chapter 86 which describes birth and boon of Andhaka.
Diti makes Kasyapa happy and thus Kasyapa offers the boon:
VaishampAyana said Long ago, after her sons were slain by Vishnu, the splendorous Vishnu, Diti worshipped Kashyapa, the son of Marichi, by doing penance. O the one of bharata race! Diti performed penance, appropriate to various times and served, obeyed and treated the sage sweetly. Then Kashyapa was pleased. The one with the wealth of penance (Kashyapa) told her: O good one! I am pleased with you. O the one who is fixed in penance! Ask for a boon. O lord! O the best among those who observe dharma! The devas killed my sons. I desire to have a highly valiant son who cannot be killed by the devas. O the daughter of DakSha! O goddess! You shall have a son who cannot be killed by the deva-s. O the one having eyes beautiful as lotus! There is no doubt in this, but leaving Rudra, the true lord of deva-s (except Shiva, no other deva will be able to kill him), because I have no power on him. Hence your son shall always protect himself from him (lord Shiva). (VaishampAyana said: O Janamejaya!) Saying this, kashyapa, who speaks the truth, touched the belly of goddess (diti ) with his finger. Then she delivered a son. O the son of kuru! (janamejaya! vaishampAyana continued) He had one thousand arms, one thousand heads, two thousand eyes and that many number of legs. O the one of bharata race! Even though he was not blind, he behaved as if he was blind. Then those who reside there, called him by the name Andhaka.
Then Andhaka starts tormenting everyone and Devas also:
The one with wicked intellect (andhaka) broke the trees in the forest and destroyed gardens. He took away by force, the horses, sons of uchchaiHshravaH (divine horse) from heaven. O the one of bharata race! (O Janamejaya! VaishampAyana continued) (Andhaka), proud due to the boon, took away forcefully, the divine elephants, sons of the elephants of directions, as the deva-s were watching. The wicked soul, the thorn of deva-s, created obstacles to those who satisfy the deva-s with sacrifices and penance.
Then the Devas together thought of slaying him:
O The best among those who observe dharma! (Janamejaya! VaishampAyana continued) All the sages, who worship brahma, distressed by andhaka, together thought about slaying him. Among them, bR^ihaspati, the one with intellect said: The death of this demon cannot happen in any way, except by rudra (shiva). While giving the boon, Kashyapa said as follows: I am not capable of protecting him from rudra (shiva). Let us think about that way, by which, the eternal Sharva (Shiva), the auspicious lord will know about the distress caused for all beings. The lord of the world (Shiva), the lord who knows about the distress, the ultimate refuge of the good, will wipe away the tears. The lord of deva-s, the preceptor of the world, Bhava (Shiva) has a pledge to protect the good, especially the brahmins, from the bad. Let all of us seek the refuge of nArada, the best among the twice born. He will find a way, since he is the companion of Bhava (Shiva).
Then Narada on the way meets Andhaka and he asks about flowers of Mandara mountain. Narada tells about it's importancd then he begins to act importantly. This is described in chapter 87 of the same Parva.
Many lions (kings of animals) roamed on the mountain. There were many herds of deer on the mountain. The demon, proud of his power, told the mandara mountain, which appeared in the body form. (Andhaka said) You know that I cannot be killed due to the power of the boon given by my father. All the three worlds, along with the moving and unmoving beings are under my control. O mountain! No one ever wishes to fight me due to fear. O great mountain! I am aware of the forest of pArijAta (mandAra, Erythrina Indica) which exist on your summit, decorated with flowers that provides all desires. That forest is the best jewel here.
Then he starts behaving more arrogantly. He starts crushing mountains. Lord Shiva reaches there to slay Andhaka.
When Andhaka spoke thus, the great lord (Shiva), holding a spear, mounted on a bull, arrived there for killing Andhaka. As Hara, the lord of the group of bhUtas, having intellect, became angry, along with his followers, all the three world trembled. The rivers flowed upwards. Water caught fire and burned brightly. Due to the splendor of Shiva, the entire directions caught fire.
Then Lord Shiva finally killed Andhaka:
The lord released his spear, having splendor as blazing fire. That terrible spear struck on the chest of andhaka. It reduced the terrible demon Andhaka, a thorn for the good people, to ashes. Then the groups of devas, all the sages having wealth of penance praised and pleased ShaNkara (Shiva) as he killed the enemy of the world.