- What is $\lambda$?
$\lambda$ is the wavelength of the standing wave. You can look at the equation $n_x^2+n_y^2+n_z^2=\frac{4L^2}{\lambda^2}$ as a condition that tells you what values of $\lambda$ are allowed to exist in your cavity. For each triple of integers $(n_x, n_y, n_z)$, you calculate $\lambda$, and that wavelength will fit in the box. Of course, there are degeneracies; some triples will lead to the same value of $\lambda$, such as $(1, 0, 0)$ and $(0, 1, 0)$. We can say that the wave corresponding to this value of $\lambda$ is an allowed mode in the cavity.
- Why look at the volume?
First, the idea of looking at volumes and areas in $n$-space instead of counting discrete points is only valid in the large $L$ limit, when you can think of the allowed values of $\lambda$ as forming a continuum. (Your head may be spinning that we started off by pointing out that only discrete values of $\lambda$ are allowed, and then immediately make an approximation that lets $\lambda$ be continuous. Just stick with it, it will pay off.)
Second, the language on the website you linked to is not very precise. If you want to satisfy the equation $n_x^2+n_y^2+n_z^2=4 L^2 / \lambda^2$, then in $n$-space you are looking for a spherical surface with radius $2 L / \lambda$.
The calculation that is being done (even though this is not explained well) is to say that there is a cap on the energy that a mode can have. Saying there is a maximum energy is equivalent to saying that there is a minimum wavelength (because $E=h f = h c / \lambda$), which means in $n$-space there is a maximum radius. The volume of a sphere with that radius counts the number of modes that do not exceed this maximum energy. Actually you have to multiply the volume of the sphere by $\frac{1}{8}$ to account for the condition that $n_x, n_y, n_z$ are all non-negative, and by $2$ to account for the fact that a photon has two polarization states.