What are the sub-shells in atomic orbitals?

Answer

You can break down the shell and orbitals of an atom on several levels of detail.
First, there are the main shells of the atom, and these are numbered 1, 2, 3, 4, etc. (or like some doing it alphabetically K, L, M, N, ...).
In the periodic table you will encounter them as seven periods.

Each shell holds orbitals (sometimes called sub-shells, but they are not), and different shells have different numbers of orbitals associated with it. The higher the number of the main shell, the more orbitals it contains. In fact, the number of the main shell is equal to the number of orbitals it contains. The types of orbitals are labeled like this: s, p, d, f, and not so very important: g, h, i, etc (alphabetically after i).

So far we have:

Main shell #1
-- contains one orbital (s-orbital)

Main shell #2
-- contains two orbitals (s-orbital, and p-orbital)

Main shell #3
-- contains three orbitals (s-orbital, p-orbital, and d-orbital)

Main shell #4
-- contains four orbitals (s-orbital, p-orbital, d-orbital, and f-orbital)

etc...

Now we can further break down orbitals! The p-orbital is actually composed of three sub-orbitals and the d-orbital is composed of 5 sub-orbitals and f-orbital in 7 sub-orbitals.
So we have:

s-orbital: just a single orbital, called just the s-orbital

p-orbitals: composed of 3 sub-orbitals, called the p_x-, p_y-, and p_z-orbitals

d-orbitals: composed of 5 sub-orbitals, called the d_xy-, d_xz-, d_yz-, d_x²-y², and d_z²-orbtials

f-orbitals: composed of 7 sub-orbitals, called f_z³, f_xz², f_yz², f_xyz, f_y(3x²-y²), f_x(x²-3y²), f_y(3x²-y²).

etc...

And at last, but not least: each sub-orbital has a maximum of TWO electrons in it, only differing in electron spin number.
So the maxima per orbital are:
s-orbital: just a single orbital, max. 2 electrons
p-orbitals: composed of 3 sub-orbitals, max. 6 electrons
d-orbitals: composed of 5 sub-orbitals, max. 10 electrons (= number of transitional elements)
f-orbitals: composed of 7 sub-orbitals, max. 14 electrons (= number of lanthanides, actinides)

(Thanks to JEK, who's original answer I completed with the last paragraph and some minor add's in the first ones)