How are the components of the spin vector defined? How do we distinguish between the $x$, $y$ and $z$ spin components?
More precisely:


*

*how do we define the $z$ component? (according to what, it is the $z$?)

*for measuring the $x$ component how should we define the $x$?


As a whole, I have problems in understanding the Stern-Gerlach experiment.
 A: Spin of an elementary particles is not necessarily the result of a movement of the particle around itself i.e. around some rotation axis that passes through the particle.If there were such an axis, the projection of the spin in the plane perpendicular to that axis were zero. But, this is not the case. So, along whatever axis we would measure the spin, we would get a non-zero result.
Though, we use sometimes to "polarize" the particle, i.e. to force the spin to be oriented in some direction. For such purposes (and others) we use the Stern-Gerlach apparatus. For a non-polarized beam of particles (see what I said above), we pass the beam through the apparatus, the beam is split into a couple of thinner fascicles, each one with a definite polarization (spin direction), and we block all the superfluous fascicles.
Now, for the z axis, you can take whatever direction in the space, that you please. Just for the x and y, if they are needed, you will have to choose axes in the plane perpendicular to the z that you chose.
Now, about the Stern-Gerlach experiment, I'll tell you in brief how it works. 
Let's take a neutron. It has, beside spin, a magnetic momentum proportional to the spin. So, if we measure the magnetic momentum, we get some idea about the spin. 
The spin of the neutron is ħ/2, and its projection on whatever axis is either ħ/2, or -ħ/2. There are no intermediate values. Then the projection of the magnetic momentum on the axis that you picked will be discrete, either positive, or negative.
If you want to measure the magnetic momentum along, say, the vertical direction pointing upwards, you do a trick: create a magnetic field H pointing downwards. Bring your particle to pass through this field in the plane x, y. If the magnetic momentum points upwards, the configuration of the apparatus containing the field H as described, will deflect the particle upwards. If the magnetic momentum points downwards, the apparatus will deflect the particle downwards.(About the configuration read in Wikipedia http://en.wikipedia.org/wiki/Stern%E2%80%93Gerlach_experiment).
Thus, even if the particle are non-polarized in the beginning, it leaves the apparatus either in upwards direction or in downwards. And if you are interested to pick the upwards direction, block or ignore the other one.
But for other details, see Wikipedia.
Good luck !
A: Generally speaking, the choice of what the $z$-axis (equivalently $x,y$) is is arbitrary. You can choose any direction to be your $z$-axis, as long as you do the calculations consistently with this choice.
If the system has a priviliged direction (like that imposed by the magnetic field in the Stern-Gerlach case) that is usually choosen to be the $z$-axis. But, again, this is a matter of conventions, you could have equally have defined it as the $x$ or $y$-axis.
Once you've chosen a convention, the difference in measuring the spin along different directions is... well, that you are measuring the spin along different directions (what is the difference in measuring width and height?).
In quantum mechanics, the fact the spin operators $S_i, \,\, i=x,y,z$ corresponding to different directions do not commute with each other (you have $[S_i,S_j] = i \epsilon_{ijk} S_k$), implies that if you measure the spin along (e.g.) the $z$ direction (i.e. $S_z$) you lose information on what the spin is in the $y$ or $x$ direction.
