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