Does anyone have an answer to the question "What does the cotangent complex measure?"

Algebraic intuitions (like "homology measures how far a sequence is from being exact") are as welcome as geometric ones (like "homology detects holes"), as are intuitions which do not exactly answer the above question.

In particular: Do the degrees have a meaning? E.g. if an ideal I in a ring A is generated by a regular sequence, the cotangent complex of the quotient map A->A/I is (I/I^2)[-1]. Why does it live in degree 1?