In which I attempt to describe spin...
In physics, spin is the a general term for angular momentum (it's rotation, like that of the Earth) of a particle.* In MRI we use a slightly more specific definition.
*Here I have described orbital angular momentum.
Spin Angular Momentum
“I have not yet lost a feeling of wonder, and of delight, that this delicate motion should reside in all the things around us, revealing itself only to him who looks for it. I remember, in the winter of our first experiments, just seven years ago, looking on snow with new eyes. There the snow lay around my doorstep – great heaps of protons quietly precessing in the earth’s magnetic field. To see the world for a moment as something rich and strange is the private reward of many a discovery.” ~ Edward Mills Purcell
In MRI spin refers to the quantum mechanical property of spin angular momentum. This gives rise to magnetic properties in the particles. They don't actually spin on their axis like the Earth does but it is far easier to explain these concepts by visualising it in this way.* The Stern-Gerlach experiment shows that particles have spin. 
Spin is a fundamental property of particles.† An individual particle can have a spin of 0, ±½ or ±1.  When particles are combined their spins add so two particles with opposite spins will cancel out while two with the same spin will double their spin value and so on.
When a particle which has spin is placed in a magnetic field it will precess (i.e. rotate around) around it. It does so at a specific frequency known as the Larmor Frequency which is incredibly important for MR as we will find out next time. 
*I'm not doing a primer on quantum here - I'm sure that's already been done far better by someone else.
†This means all particles in the Standard Model have a spin quantum number.
Next week (hopefully) we will get onto the topic of how to actually make these spinning particles give us some information.
 Walther Gerlach & Otto Stern (1922). Das magnetische Moment des Silberatoms. Zeitschrift für Physik 9: 353–355. doi:10.1007/BF01326984
 Joseph P. Hornak (2010). The Basics of MRI. Online Book: < www.cis.rit.edu/htbooks/mri > [19 February 2013].
 Malcolm H. Levitt (2001). Spin Dynamics: Basic Principles of NMR Spectroscopy. Chichester: Wiley.