The following passage from Thomas Numerical PDE:
What is the meaning of two level scheme? For example, lets take $v_t + v_x = 0$. The usual scheme I have studied so far for example is
$$ \frac{ u_{k+1}^n - u_k^n }{\Delta x} + \frac{ u_k^{n+1}- u_k^n }{\Delta t } = 0$$
where $u_k^n = v(k\Delta x, n \Delta t)$ . What does Thomas mean by two-level scheme?
When Thomas refers to "levels", what he means is actually time instants or steps. The scheme you write for the simple equation would, in his notation, be a two-level scheme. In particular, here he denotes by $\mathbf{u}^n$ the vector of all space values of the approximation at time $t_n$.
A multilevel scheme on $l+1$ levels (often in literature multistep) would be a scheme that involves the time snapshots of the solutions $\mathbf{u}^{n-1}, \mathbf{u}^{n-2},\ldots \mathbf{u}^{n-l}$ to compute the value $\mathbf{u}^n$.
In any case, this is explained quite thoroughly in the book you cite, two pages before your screenshot (beginning of section 2.2.4).