First thing you need to understand is that orbitals are not actual physical things that exist. Simply put, an orbital is a function that describes the probability of finding the electron with certain energy at certain distance from the nucleus. The shapes of the orbitals are just the boundaries of space where you can find the electron 90% of the time.
Electrons behave both as particles and waves. Their wave behavior is described by what is called a wave function. It has two components - radial and angular. The radial component depends on the distance from the nucleus while the angular depends on the direction. It is the angular component that shapes the orbitals. Just as it is, the wave function does not have any physical meaning, but it's square is proportional to the probability of finding the electron in a particular region of space around the nucleus.
The x, y and z axis simply refer to the Cartesian coordinate system and can describe points in 3D space. Imagine three lines where each of them is perpendicular to the other two. It doesn't matter which one you name x, y or z, this is all arbitrary.
The electrons around the nucleus are not really ordered. The terms shell, sub-shell, orbital are just a way of describing different energy levels and probabilities of finding electrons around the nucleus.
There are four quantum numbers that describe electrons in atoms - principal, azimuthal, magnetic and spin.
The principal quantum number (n) can take integer values from 1 to infinity. It describes the different energy levels with increasing distance from the nucleus. As n gets bigger, orbitals within that "shell" also increase in size.
The azimuthal quantum number (l) can take values from 0 to n-1. It describes the type of orbital the electron resides in. It is also call angular momentum quantum number and as I've said above, the angular wave function is responsible for the shape of the orbitals. Makes sense? For each value of l, there is a different type of orbital - 0=s, 1=p, 2=d, 4=f, 5=g (not really seen occupied in the ground state of any element), and these can go on alphabetically.
The magnetic quantum number (ml) describes the direction of the orbital in space with respect to the three Cartesian axis - x, y and z. It can take values from -l to +l. For s orbital, ml can only be 0 and so there is only one s orbital that can exist for a particular principal quantum number. For p orbitals, ml can be -1, 0 and +1 and so there are three p orbitals that can exist - each along different axis - px, py and pz. The axis labeling get more complicated for d and f orbitals.
The spin quantum number (ms) can either be +1/2 or -1/2 and simply describes the direction of the electron's intrinsic angular momentum.
For n=1, l can only be 0 and so there is only an s orbital in that level, referred to as 1s. For n=2, l=0,1 and there can be s and p-types orbitals. For n=3, l=0,1,2 and so you can have s, p and d orbitals in the third shell.
When you're filling up shells with electrons, you go from the lowest energy level to the highest and this can easily be determined by summing up n+l which would give you the relative energies of orbitals. When n+l is the same for two orbitals, the one with the lower n fills up first. One orbital can house 2 electrons which must have opposite spins. There can never be two electrons with the same four quantum numbers in a system (Pauli exclusion principle).