NMR theory can be tough. Luckily, most NMR theory can be visualized to aid you in understanding. The animations below show precession with a 180 degree pulse, J-coupling and T1/T2 relaxation.
Precession with 180 degree pulse
After a 90 degree pulse on x, the magnetisation is at the y-axis and starts to precess with the Larmor frequency. After some time t a 180 degree pulse on y is given, rotating the magnetisation. Then the magnetisation continues to precess and after the same time period as before the pulse (at 2t), it ends up where it started, as if no precession had taken place.
When spin A of a nucleus is coupled to a spin B, it can 'sense' the position of that spin in the B0 field. If spin B is with the field, spin A resonates at a frequency different from when spin B is against the field. The population of A spins is therefore split into two.
The red magnetisation vector represents one of these sub-populations and runs faster than the rotating frame; the blue one represents the other population and runs more slowly. The frequency difference between the vectors is the J-coupling. In 1/J seconds, each vector turns 180 degrees and they meet at the -Y axis.
The T2 relaxation causes the horizontal (xy) magnetisation to decay. T1 relaxation re-establishes the z-magnetisation. Note that T1 relaxation is often slower than T2 relaxation.
When a nucleus experiences two or more states with different chemical properties, it is in exchange. The NMR signal(s) of the nucleus depend on the exchange rate (k), relative to the difference in the chemical shifts of the states. The animations show how the spectrum changes with k for a two state exchange with a chemical shift difference of 50 rad s-1. The natural line width was neglected in this animation. At 50/sqr(2), the two peaks show coalescence.
T1 relaxation times can be measured by inverting z-magnetisation and following the recovery. After a delay, a 90-degree pulse is applied to 'read' the amount of z-magnetisation. By increasing the delay, the recovery rate (= 1/T1) can be determined.