One should reflect about what "observable" really means. In this case you can take the word literally: It's something you can observe in the real world, i.e., and for physicists this even means you can quantify it (the more precise the better). The spin of particles is a quite difficult concept, because it has no classical analogy. In the quantumtheoretical formalism it is defined in a quite abstract way, involving pretty advanced mathematics (group representation theory).
Physically, however, at least for charged particles, it leads to something very concrete: The particle, e.g., an electron has a magnetic dipole moment. So you can think of the electron (in a rough way) as a charged point particle which is at the same time a tiny premanent magnet. Performing an experiment with a single electron in order to measure it's dipole moment is not so easy, because usually its motion in electromagnetic fields is dominated by the charge and the electric field. So in 1923 Stern and Gerlach performed an experiment with neutral silver atoms. It was already then known that the silver atom is built in a way that to a good accuracy its magnet moment is that of its single valence electron, but as a whole the silver atom is electrically neutral. So the idea was to measure the magnetic moment of silver atoms by running them through an inhomogeneous magnetic field, which has a large nearly constant component in one direction (usually taken as the ##z## direction of a coordinate system) and a piece varying rapidly in space. The latter component leads to a force acting on the silver atom (as known from classical physics!). In the classical picture, the constant component of the magnetic field leads to a rapid rotation of the components of the dipole moment perpendicular to the magnetic field's direction, i.e., the ##z## direction. Thus, for the much slower motion of the silver atom, the force according to these perpendicular components averages to 0, and what's left is the motion of a dipole magnetic with the dipole directed along the ##z## direction. This means the silver atom is reflected by the force due to the inhomogeneous magnetic field to the one or the other direction perpendicular direction due to the ##z##-component of the dipole moment. In a classical picture this dipole moment can have any value, and thus one expects a broad spot when measuring many silver atoms running through this Stern-Gerlach apparatus, but what came out in this very important experiment was totally different! The beam of silver atoms split into two distinct lines registered on a photographic plate (which worked, by the way, only thanks to the cheap cigars smoked by Stern and Gerlach during their experiment, helping to better the contrast of these "photographs" due to a large amount of sulfur contained in the cigar smoke ;-)). This finding implied that the spin-z component is quantized, i.e., it can take only two values. In 1923 the correct quantum theory of spin and the related magnetic moment was not known and thus the experiment not fully understood from our modern point of view. Funnily enough two wrong implications of the then known Bohr-Sommerfeld model of atoms canceled out and lead to the right prediction for the "quantization of direction", as the phenomenon was dubbed then. Nowadays we know that within modern QT the explanation is a bit more abstract, because it is due to the half-integer spin of the electron (it has spin 1/2) and the socalled gyrofactor which is close to 2 for an electron (the latter is a relativistic effect; a naive non-relativistic treatment leads to a prediction of a gyrofactor of 1, but that's another story).
The very amusing story about the Stern-Gerlach experiment can be found in a nice Article by Herschbach et al in Physics Today:
http://scitation.aip.org/content/aip/magazine/physicstoday/article/56/12/10.1063/1.1650229