a bit of lay-hn-reader explanation from a chemistry/math major-now-dev (my physics might be a bit wrong, apologies in advance).

These diagrams show the probability density of the electrons around a hydrogen nucleus, which is the simplest (and a pretty good in general) model for how electrons live around atom nuclei. The more dots, the denser the probability, aka: how likely or not one might find an electron in this particular position.

In the upper right corner, there's the psi(n, l, m) selector which lets you pick the geometry.

n is the "principal quantum number" which corresponds to "the gross energy level/frequency" of electron. The way to think about this (I think) is this: If you are plucking a string on a guitar, or play a wind instrument, the more nodes that it has, the higher the energy of the vibration. Similarly for electrons around atoms. As you pick diagrams with a higher n, you'll see more nodes (internal regions with zero density) in the distribution. These are also higher energy states. Generally, if you look carefully you should be able to find (n - 1) surfaces, though for the (n, 0, 0) diagrams some of these node surfaces are tiny spheres close to the nucleus, so you might not see them.

l is the angular quantum number. This number determines how many of those nodal surfaces are "not spherical". So in a (n, 1, X) diagram, you should eventually see a plane cutting through if you play around with the orientation; In an (n, 2, x) you should see two intersecting planes cutting through, or in some cases a cone (more on that later).

m is the magnetic quantum number, and presumes that the atom is sitting in a nonzero magnetic field, and selects for different energies that relative orientations in that magnetic field have. This splits the different possibilities based on direction relative to magnetic field, and not curve qualities (number of nodes; shape of nodes).

There's another quantum number, which is the "spin quantum number" that has to do with the Pauli Exclusion principle, that two electrons can share an orbit simultaneously. This doesn't really change the shape of the orbital, so I presume that's why it's not there.

(1, 0, 0) is possible, but probably not shown because it's boring.

As for why you could have a "plane" or a "cone"; the display coordinate systems are somewhat arbitrary, and as with most quantum mechanics, "reality" is actually a weighted linear sum (superposition) of all of these possibilities; so a "plane" and a "cone" are roughly equivalently "surfaces", but the cone is a linear combination of a bunch of planes rotated around a line but is selected because it's a convenient and easy basis component with the other "planes" to generate coverage of the vector space of all possibilities. To really butcher the explanation: It turns out that you have to play that "rotate trick" because the space of "all possible probability distributions" has a fixed dimension, and you run out of ways to chop up three dimensional spaces with planes, so you have to mash them together to get correct coverage of the space of distributions.

How this corresponds to the periodic table. The S block (left side) elements are mostly filling their (row, 0, 0) orbitals, then P block (right side) elements are filling their (row - 1, 1, _) orbitals. The transition metals are filling their (row - 2, 2, _) orbitals, and the inner transition metals are filling their (row - 3, 3, _) orbitals . Although it seems elegant, reasoning for the "row - X" and not "row" is a bit complicated, empirical and not theoretical, and if you'd like to understand why, look up "aufbau principle".