The basis of a of a matrix is the set of the smallest amount of linearly independent vectors that we can choose that cover the entire space based on their span.
This implies that a non-zero subspace has infinitely many different bases, but they all contain the same number of vectors.
The number of vectors in any basis of a subspace V is called the dimension of V, and is written dim V.
This is also equivalent to the reduced form of the column space. It is from my understanding the non-redundant, reduced form of the column space of a matrix.
Standard Basis
The standard basis of R^n is effectively the set of vectors with the axes being 1, and the rest 0, or the identity matrix of that space.