What does it really mean that particle has a spin of up/down? And how is spin actually meassured? I was reading a physics article (related to the recent discovery of a particle that could be the Higgs boson) and the article was discussing electron spin and how spin can only be either up or down. I've always found this confusing – I know the directions 'up' and 'down' can be seen as arbitrary as they are relativistic in space. So, unless spin orients itself depending on the nearest gravity source, spin in this context must mean something different than in ordinary English. 
I'm also confused by the claim in the article that you can have right or 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 the 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 as to 1. why spin is up or down and 2. why it can come off as left and right when measured).
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
 A: Your confusion probably arises not from the technical details of spin measurement, but from the peculiar nature of quantum mechanics.
The spin state of an electron can be arbitrarily 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 set of basis vectors of this space. In other words, any spin state may be written as a linear combination of up and down states (or left and right states). Designating up and down states as the basis is analogous to choosing a 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 the quantum state. Whatever the initial orientation, if you measure spin along the z-axis, the outcome can only be up and down, with a certain probability. Now since a left state tilts neither upward or downward, it is a natural possibility that each outcome is 50%.
A: To answer the "how is spin measured" part of the question, look at the Stern Gerlach experiment   where particles are deflected showing instead of a uniform distribution of angular momentum, deflection either up or down. This is a direct measurement that in principle could be carried out on individual particles.
A second way to establish spins/angular momentum of  atomic energy levels is to solve the quantum mechanical equation and fit the observed absorption or emission spectrum of the atoms. The fit will give the spin/angular momentum appropriate from the solution.
A third way, mainly used in high energy particle physics is to study the angular distributions/correlations  of decay products of the resonances observed.In this link the study of the spin of the recently discovered Higgs boson is explained on page 20, angular correlations are used for this purpose . The advantage of having decay modes allow also exclusions of spins not compatible with what has been observed.
