The van der Waals equation is represented as such:
$$\left(P + \frac{an^2}{V^2}\right)(V-nb) = nRT$$
Here, $P$ is the pressure of the container and $V$ is indeed the volume of the container. All the symbols that come from the original ideal gas law maintain their definitions. The best way to conceptualize the van der Waals equation is as a perturbation of the ideal gas law that brings it closer to real gas behavior. Let's start with the ideal gas law:
$$PV = nRT$$
This equation simply states that pressure and volume are inversely proportional to each other at a constant number of molecules $n$ and a constant temperature $T$; that their product is directly proportional to number of molecules $n$ and absolute temperature $T$; and that these parameters are the only considerations to model gases' behaviors. It is not simply that the equation conforms to evidence from the world (empirical) but also that it applies in the fictional but reasonably approximate case that molecules 1) do not interact and 2) do not occupy space. It is often more useful to conceptualize an ideal gas as a gas where gas particles pass through each other; this is equivalent both to excluding particle interactions and to allowing each gas particle the entire volume of the container (because nothing gets in the way). If gas particles all passed through each other, then the ideal gas law would be true, but alas this is not the case.
Since this is not the case, we need to add parameters that account for particle interactions. An entirely new equation is not necessary though: instead of starting from scratch, we can add a term to the pressure and volume variables that corrects for the existence of particle interactions. It is simplest to demonstrate with the case of the $(V-nb)$ term:
Let's say that particles do not interact except they collide with each other and bounce off elastically like billiard balls. The volume term in the ideal gas law $V$ stands for the volume of the entire container; however, if two molecules simply collide with each other, then for every molecule added to the container, there is a molecule's volume less space for every other molecule to be. So, while we still see the volume of the container as $V$, from the molecule's perspective there isn't a full $V$ volume worth of space to go, because other molecules are in the way.
For an analogy, if you walk into an empty room, you can navigate every square foot of the room easily. However, if you invite a bunch of friends over, suddenly there is much less space to navigate even though the room did not literally shrink in size—you effectively have less space to move. We'll call the amount of space that the molecule can actually move "effective volume" or $V_\text{eff}$.
We expect for every additional molecule we add that the volume $V$ stays the same, but the effective volume $V_\text{eff}$ decreases a little bit. So, our effective volume should be 1) linearly related to the number of molecules proportional to the molecule's volume and 2) smaller than the original volume. We can use this to make an equation, where $b$ depends on the volume of the molecule:
$$V_\text{eff} = V - nb$$
A similar argument is made to account for molecular attractions with the pressure parameter, but this is much more complex and includes a squared volume term for complicated reasons. Regardless, we can construct an effective pressure $P_\text{eff}$ which is just the original pressure term with an additional correction for particle interaction included. If we let:
$$P_\text{eff} = P + \frac{an^2}{V^2}$$
and
$$V_\text{eff} = V - nb$$
and simply let $a$ and $b$ be corrections to the original pressures and volumes, then we get the van der Waals equation looking like this:
$$P_\text{eff}V_\text{eff} = nRT$$
which looks exactly like the ideal gas law, but corrected for additional factors.
