Isaac and his mother live in a simple home. Isaac would play with his toys while his mother watched Christian broadcasts on the television. One day, Isaac's mother hears the voice of God. He tells her that Isaac is filled with sin, so she takes away everything that she deems sinful (such as his toys, his pictures, and even his clothes) to save him. Soon after, God calls out to her again and says Isaac is still sinful, so she separates Isaac from the outside world, locking him in his own room.
God again contacts Isaac's mother and says she has done all that he commands, but still questions her faith, and demands a sacrifice to prove her faith (similar to the Biblical Binding of Isaac by his father Abraham). Isaac looks through a crack in his door and sees his mother coming towards his room with a butcher knife. He panics, but finds a trap door in his room under a rug and, without hesitation, jumps into it to save himself. As he traverses his basement, he will find various items to help him face various creatures, long lost siblings, his own fears, and finally his mother.
He has a diary, which he writes in when he dies and also a cat that he gives all of his material goods (items you pick up before you die) to.
He also seems to have a pretty rough life, as shown in Isaac's thoughts between levels. Some thoughts include friends that constantly pull terrible pranks on him and seeing him get close to his mother, only to be kicked aside by her.
Unlockable characters in The Binding of Isaac seem to all be Isaac, with different items on. Isaac wearing an eye patch is Cain, Isaac wearing a wig has him cross-dressing as Magdalene, and wearing a fez hat has him play as Judas. Each has different stats and starting items.
It is heavily implied in the true ending of the game (at least until the Wrath of the Lamb expansion is released) Isaac may be the Blue Baby and eventually suffocated inside of a chest while trying to hide from his mother.