Newer Books

Matrix Analysis and Applied Linear Algebra by Meyer is very well written with clear cut examples and exercises. I think this would make an excellent first course.

I agree also that Axler's books is a great text for the more mature.

Classics

Finite-Dimensional Vector Spaces by P. R. Halmos is an absolute essential for the budding mathematician in my opinion. This is because of the exercises (My recommendation: solve all of them).

As mentioned above Linear Algebra (2nd Edition) by Kenneth M Hoffman and Ray Kunze. This may be my favorite text because of its volume of content.

More Advanced

Advanced Linear Algebra by Steven Roman

Matrix Analysis

Matrix Analysis and Topics in Matrix Analysis by Roger A. Horn and Charles R. Johnson

Matrix Analysis by Rajendra Bhatia