In The Fellowship of the Ring, when Gandalf and his team enter the dwarf caves, they come across Balin's grave. Why did Balin leave Erebor and end up there? Also, how did he die?
The answer to the first part is in the Lord of the Rings chapter The Council of Elrond:
'It is now many years ago,' said Glóin, 'that a shadow of disquiet fell upon our people. Whence it came we did not at first perceive. Words began to be whispered in secret: it was said that we were hemmed in a narrow place, and that greater wealth and splendour would be found in a wider world. Some spoke of Moria: the mighty works of our fathers that are called in our own tongue Khazad-dûm; and they declared that now at last we had the power and numbers to return.'
Glóin sighed. 'Moria! Moria! Wonder of the Northern world! Too deep we delved there, and woke the nameless fear. Long have its vast mansions lain empty since the children of Durin fled. But now we spoke of it again with longing, and yet with dread; for no dwarf has dared to pass the doors of Khazad-dûm for many lives of kings, save Thrór only, and he perished. At last, however, Balin listened to the whispers, and resolved to go; and though Dáin did not give leave willingly, he took with him Ori and Óin and many of our folk, and they went away south.'
And as for his death, this in in the chapter The Bridge of Khazad-dûm:
'I fear he had ill tidings to record in a fair hand,' said Gandalf. 'The first clear word is sorrow, but the rest of the line is lost, unless it ends in estre. Yes, it must be yestre followed by day being the tenth of novembre Balin lord of Moria fell in Dimrill Dale. He went alone to look in Mirrormere. An orc shot him from behind a stone.'