What does it really mean that particle has a spin of up/down? And how is spin actually meassured?

Question

I been reading some physics articles (related to the recent discovery of the particle that could be a Higgs boson) posted online and it was talking about electron spin and how it can only have values of either up or down and that always confused me. I know directions of up and down are really arbitrary because in just space there be no up and down. So unless spin orients itself depending on the nearest gravity source, spin of up/down must mean something different then in ordinary English language.

I also been confused by the fact that the article claims that you can have right/left spin as a result of a spin measurement:

Suppose you do measure an electron as spin up, and then try to measure the left-right spin. Common sense would tell you that that number would be zero, since you know that the electron is spin up, not left or right, but I warned you about common sense before. It turns out that a) half the time you'll measure the electron to be left and the other half you'll get right, and b) whether it's left or right is completely random.

I probably would understand what is meant by up/down spin if I knew how a spin of a particle is actually measured. (I think i could handle a detailed and precise explanation but a crude explanation will suffice if it gives insight to why spin is up or down and why it can come off as left and right in measurements)

Is up down just a name given by physicists to two different types of spins? Or does it have something to do with the actual directions?

Related

What is spin as related to particles

Answer 1

Your confusion probably arises not from the technical details of spin measurement, but the peculiar nature of quantum mechanics.

The spin state of electron can be arbitrary aligned, so there are infinite possible spin states, not just up and down. But all these states live in a 2-dimensional vector space, and up and down states are one sets of basis vectors of this space. In other words, any spin state may be written as linear combination of up and down states (or left and right states). Designating up and down states as the basis is analogous to choosing coordinate system; they are arbitrary and do not establish a preferential orientation in space.

Another peculiar thing about quantum physics is the measurement induced "collapse" of quantum state. Whatever the initial orientation, if you measure spin along z axis, the outcome can only be up and down, with certain probability. Now since a left state tilts neither upward or downward, it is naturally the possibility of each outcome is 50%.