Certain nuclei exist in specific nuclear spin states when exposed to an external magnetic field. In NMR we can use this phenomenon to observe transitions between these spin states. These transitions are specific to a particular nucleus and also to the chemical environment of the the nucleus in question. The basic principles of NMR are explained in this video and then further expanded here.

The vector model is a way of visualising some aspects of NMR by following the magnetisation vector in Cartesian space. This is explained in the lectures of James Keeler which can be found both as a video and a pdf. This model is a non-quantum mechanical description of the result of the application of radiofrequency pulses. It involves similar geometric explanations and allows for easy understanding of the basic principles of pulse sequences. Although the vector model can only be applied to uncoupled spins it does enable the demonstration of principles such the pulse acquire mechanism by which the NMR signal is generated and also the mode of action of the spin echo.

The product operator formalism is rigorous quantum mechanical way of predicting evolution of the bulk magnetisation during pulse sequences. It uses clear operators that have clear physical meanings that are represented as geometric rotations but allows the prediction of many effect that are not possible using the vector model. Product operators are explained in detail in this lecture, its follow-up and also as a PDF.