Inner product of columns of a matrix

You have determined $A^*A$, or, alternatively, you know $A$ up to pre-multiplication by a unitary matrix $U$. So you know the $R$ factor of its QR factorization, and the factors $\Sigma$ and $V$ of its SVD. In particular, among other things, you know singular values and right singular vectors, the sign of its determinant, its rank, its kernel, its Euclidean and Frobenius norm.

A more geometric supplement to Federico's answer: if you think of columns as vectors, then knowing the inner products determines the set of vectors up to rigid motion, so any geometric information is determined by the inner products (including the volume of the simplex they span [determinant], singular values [the semi-axes of the ellipsoid which is the image of the unit ball by your matrix], the areas of the faces of the simplex determined by them, etc, etc.