1s2 = 2
2s2 + 2p6 = 8
3s2 + 3p6 = 8
4s2 + 4p6 + 3d10 = 18
5s2 + 5p6 + 4d10 = 18
6s2 + 6p6 + 5d10 + 4f14 = 32
7s2 + 7p6 + 6d10 + 5f14 = 32

its all about the electron shells!

each orbital can contain 2 electrons (with oposite spins denoted +1/2 or -1/2)

the s shell has 1 orbital = 2 e-
the p shell has 3 = 6 e-
the d shell has 5 = 10 e-
the f shell has 7 = 14 e-
and the g shell will have 9 orbitals even though they would be synethic/transuranic elements. I'm not sure they would call it the g, but that sounds right for some reason.

the very clever Russian man, Mr Mendelev, invented the concept of a 'period' in his periodic table.

Deryk is of course right about the how many electrons there are in those filled-up orbitals.

For a deeper explanation of where all those numbers come from, you need to look at quantum mechanics. It will tell you stuff about "degeneracy" and quantum numbers - spin, orbital & spin angular momentum, and other gubbins.

Then do you have a brief summary (as short as possible) about that?

It is easier to understand with pictures of the orbitals and equations -
it would probably be better if you read some of a first-year undergraduate text book about quantum mechanics ....

But I will try!

First of all, we can think of the electron being a "vibrating" sphere of negative charge around the nucleus.
The vibrations are "quantized" into "spherical harmonics"... this is analagous to the vibrating string (eg in a violin, a piano, or guitar) - the string is fixed at either end and so the vibrations are quantised into "harmonics" (modes", of a fundamental frequency f, then 2f, 3f, 4f...
The spherical harmonics the electron can vibrate around the nucleus are also quantised like this - we say this corresponds to principle quantum numbers 1, 2, 3, 4..... and these correspond to the spherical "s" orbitals 1s, 2s, 3s, 4s....

The electrons can also vibrate in ways that have orbital angular momentum (l) This leads to different shaped orbitals....p orbitals, d orbitals, f orbitals....... (each type corresponds to more angular momentum).... because this momentum is a vector, the various combinations of up and down (reinforcing or cancelling out) means that you get 3 p orbital, 5 d orbitals, 7 f orbitals...(ie there's 3 unique ways, of equivalent energy, of laying out the angular momentum vectors of p electreons, and 5 ways for d... etc etc)

Lastly, electrons have a magnetic spin momenet of their own (you can think of this as that they spin on their own axis, either clockways or anticlockwise).... so in each orbital, you can fit two electrons, one up and one down.....

There are pages of equations which will define all these things more rigorously for you, if you are interested!
But most simply, the numbers come from the number of ways you can arrange electrons in vibrating "spherical harmonics" around the nucleus,
taking account of their spin (up or down) and the orientations of their angular momentum.

And thats about as simple and brief as I've ever heard. nice work feline

But still a question. Why 3p, 5d, 7f?

"p", "d" and "f" are just labels that we have decided to call some orbitals.

The letters come from spectroscopy.... they refer to series of visible spectral lines, which had been called the "principle series", the "diffuse series" and the "fundamental series".... it turned out that these series corresponded to an electron transition to the p orbital, the d orbital, and the f orbital, respectively.

If we say that electrons are in a "p orbital", what we mean is that they have 1 unit of quantised angular momentum.

Likewise, if we say that electrons are in a "d orbital", what we mean is that they have 2 units of quantised angular momentum.
For an f orbital 3 units of angular momentum.

If you examine the Schoedinger equation for an electron that's bound on a spherical harmonic around a nucleus, and that has 1 unit of angular momentum, you find that there are THREE equivalent solutions (ie, solutions that have equal energy - we call this "degeneracy")
Physically, you can think of it like this
you have 2 electrons in the orbital (one is spin up, one is spin down). They both have one unit of angular momentum.
However, angular momentum is a vector quantity (ie, it points in a particular direction) There are 3 different directions that the two electrons in the orbital could point their angular momentum vectors, and still have the same energy. (I can't draw you a picture here, sorry! which would help!)
So, this means, there are THREE p orbitals in each quantum level. (in other words, in each spherical harmonic, if the electrons have 1 unit of angular momentum, there are THREE equivalent (degenerate = equal energy) ways that they could orbit the nucleus.
The real solutions of the schroedinger equation look like the 3 "dumbells" we call "p orbitals"

If the electrons had TWO units of angular momentum,
we call this "d orbitals", and it turns out there are FIVE ways you could arrange the angular momentum vectors that are degenerate. So there are FIVE d orbitals.

And if they have THREE units of angular momentum, we call this "f orbitals", and it turns out there's SEVEN ways you could point their angular momentum vectors that are degenerate - so you can have 7 f orbitals.

(This could go on and on .... if electrons had 4 units of angular momentum, we'd call it a g orbital... and I don't know what the solutions are, but you could calculate how many equivalent ways there were to arrange the vectors, and this would tell you how many g orbitals there'd be)

BUT - the more angular momentum an electron has, the more energy it would have - so we generally don't see electrons with 4 units of angular momentum (g orbitals) for the ground states of the known elements.

ALSO you have to bear in mind that there's not really any such thing as orbitals anyways lol lol
It is just an approximation to treat the orbitals as all seperate things,
In reality, you should have a single Schroedinger equation for the wavefunction for the ENTIRE ATOM - the nucleus and all the electrons at once.
However, since we can't solve equations of motion for 3 bodies or more, we just make this serparate "orbital approximation" instead.

It works well for small atoms, but at the bottom of the periodic table, it's rubbish lol
(this is because the repulsions between all the dozens of electrons are much bigger than the attractions between them and the nucleus)
You can tell its rubbish my looking at spectra of heavy elements.

Feline, you've got my point incorrectly. I'm asking why s starts from 1, p from 2, d from 3, f from 4, g from 5, but they don't fill up itselves on the same shell? (I think that's because of energy levels?)
Why 2, 6, 10, 14 not 4, 8, 12, 16?
Why 2, 4, 6, 8 (the starting shell of a new orbital) not 1, 3, 5, 7?

I think it is just the way the solutions of the Shroedinger equation work out?

Try not to think of the concept of "shells" - it doesn't really correspond to any physical reality!

Each "shell" corresponds to an increase in the "principle quantum number" - I think the closest "physical" thing this corresponds to is the number of "nodes" (= zero crossing points) in the shape of the wavefunction.

So if you have no nodes in the wavefunction, the only shape that will fit as a spherical harmonic is the 1s shape (just a sphere). The orbital angular momentum quantum number ("l") is zero in that case.

the next set of solutions (wavefunction shapes) correspond to 2s and 2p - the 2s has a spherical node in it (ie, the phase of the wavefunction flips inside out radially),
and the 2p orbitals have angular momentum = 1

There's no valid solution for the schroedinger equation with principle quantum number = 2 but l = 2 (in other words, there's no 2d or 2f orbitals)

I don't know if this really "explains" it for you? If you get a textbook and look at the mathematics, you will see the solutions to the schroedinger equations.... but it is very complicated.

Maybe a better way to think of it is simply "because those are the only patterns which will fit" ? Others are not the right shapes!

For an analogy, try to find example pictures of the modes of vibrations of some musical instruments look first at strings and pipes ... then look at the modes of vibration of stetched membrance (eg a drum skin, or a cymbal) .... you will see some wierd shapes ..... these are just the shapes that fit those objects...... it is the same (sort of) with electrons and atoms.

On the other hand, all these things are just made up mathematical approximations lol
They are not always really real. (Both literally philosophically, and in the mathematical sense ) lol
Most of the equations are approximations in the first place,
and even the approximations have no exact mathematical solutions, and you have to use numerical iterations to try and get a fit....
....then you compare these results to your actual empirical ones (eg from spectroscopy) and see how good (or bad) they are.....

Nearer already, but still don't get to the total answer.

Why doesn't the subshell with quantum number 2 come with 4 electrons but 6 electrons?