What is range of a matrix?

In the simplest terms, the range of a matrix is literally the "range" of it. The crux of this definition is essentially

Given some matrix $A$, which vectors can be expressed as a linear combination of its columns?

Range (another word for column space) is what is meant by this. If you give me some matrix $A$ that is $m \times n$, the column space is the set of all vectors such that there exists $a_1, a_2, ...., a_n$ so that $a_1A_1 + a_2A_2 + ... a_nA_n = v$ for some vector $v$.

$$\begin{bmatrix}1 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix} \begin{bmatrix}a_1 \\ a_2 \\ a_3\end{bmatrix}= \begin{bmatrix}5 \\ 5 \\ 5\end{bmatrix}$$ Then $v$ is in the range of $A$ since $a_1 = a_2 = a_3 = 5$. A better example is when it's not, like: $$\begin{bmatrix}1 & 0 & 3\\ 1 & 1 & 2 \\ 0 & 0 & 0\end{bmatrix}\begin{bmatrix}a_1 \\ a_2 \\ a_3\end{bmatrix} = \begin{bmatrix}5 \\ 5 \\ 5\end{bmatrix}$$ Now it's not... since no $a_1, a_2, a_3$ will satisfy the condition that $v$ is a linear combination of the columns of $A$...I mean, we will always have $0$ in the third entry of any linear combination!

From this definition, the null space of $A$ is the set of all vectors such that $Av = 0$. Obviously $v = [0, 0, 0, ..., 0]$ is part of the null space, so it is always non-empty.

The rank of the matrix is related to the range. It denotes how many columns of $A$ are actually "relevant" in determining its range. You may think that removing a column from a matrix will dramatically affect which vectors it can reach, but consider: $$\begin{bmatrix}1 & 2 & 0\\ 1 & 2 & 0 \\ 1 & 2 & 0\end{bmatrix} \approx \begin{bmatrix}1 \\ 1 \\ 1\end{bmatrix}$$ You can try to reason (to yourself), that the left matrix can reach the same space of vectors as the right matrix (Why?)