column space
The column space of a matrix is the span of its columns, that is, every vector you can build by mixing the columns with scaling and addition. Since multiplying A by x just blends the columns of A using the entries of x as weights, the column space is precisely the set of all possible outputs A*x.
Think of it as the reach of the matrix: all the places it can possibly land you. If you are solving A*x = b, a solution exists exactly when b sits inside the column space; if b is outside the reach, no choice of x will ever get there.
Its dimension is the rank. A wide matrix whose columns only span a plane has a column space that is a plane, no matter how big the matrix looks, and its rank is the dimension of that plane.
The columns point the same way, so every output lands on one line: the matrix can only reach that line.
Column space lives in the output side; null space lives in the input side. The two together tell the whole story of what the matrix does.