Why is the detection sensitivity of NMR so low?

This can be explained most simply by treating the nuclear spins in a sample (semi-)classically—as if each nuclear spin were a tiny bar magnet:

When placed in a strong magnetic field (B₀), each spin (for example, within the hydrogen nuclei of water) will tend to align either with the magnetic field ("spin up") or against it ("spin down"):

In NMR spectroscopy, one detects the response of the bulk magnetization produced by the nuclear spins in the sample (after perturbing this magnetization from equilibrium with resonant radio-frequency (RF) radiation—the precessing magnetization then induces an oscillating signal in a pick-up coil, which is the source of the NMR spectrum).  The strength of this magnetization (M₀) is determined not only by the (extremely small) size of the magnetic moments of the nuclei in question, but also by the net difference between the number of nuclear spins aligned with—and against—B₀ (for example, if the two spin states were equally populated throughout the sample, the contributions from the spins would essentially cancel each other out, leaving no net magnetization).  This fractional population difference is known as the polarization, P.  When the spins are at thermal equilibrium with their surroundings (including B₀), the population difference is dictated by the Boltzmann distribution to give:

where N refers to the number of spins in a given state, g is the gyromagnetic ratio for the given nuclei (proportional to the magnetic moment), h is Planck’s constant divided by 2p, k_B is Boltzmann’s constant, and T is the sample temperature (in Kelvin).  Because of the weakness of nuclear magnetic moments, the difference in energy between “spin up” and “spin down” states is very small (which, by the way, explains why the resonant photons are in the low-energy RF regime of the electromagnetic spectrum) giving DE = ghB₀ << k_BT .  This fact, in turn, forces the populations of the spin states to be very nearly equal (often differing by only ~0.001% or less)—thus yielding very little net magnetization. 

Of course, one can see from the equation above that one can improve the NMR sensitivity by either raising the strength of B₀ or dramatically lowering the temperature, but there are obvious limits to how far one can go with such an approach (stronger magnets grow exponentially more expensive, and many samples do not “like” to be cooled more than a few degrees—particularly living ones).  Instead, if one could change the relative populations of the spin states by some other means (i.e., to achieve a highly non-Boltzmann distribution—as if to make the spins “colder” than the molecules which contain them), then one can dramatically enhance the NMR detection sensitivity of these molecules…