What information about a (square) matrix we earn if the inner product of its columns are known?
You have determined $A^*A$, or, alternatively, you know $A$ up to premultiplication 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 semiaxes 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.

$\begingroup$ Where can I find such relationships, say a formula for determinant of a matrix by inner product of its columns? $\endgroup$ – M. Farrokhi D. G. Jan 26 '17 at 4:38

1$\begingroup$ @M.FarrokhiD.G. The matrix of inner products is $A A^t,$ so its determinant is the square of the determinant of $A.$ For various other formulas, check out my paper Surface Area and Other Measures of Ellipsoids. $\endgroup$ – Igor Rivin Jan 26 '17 at 4:45