There are many uses for deuterium in NMR, but one is clearly dominant, and that one is of technological (instrumental) nature: to achieve the required magnetic field stability.
Since magnetic resonance is the absolutely most precise technique to measure magnetic fields, the field stability that NMR measurements require can be only achieved by incorporating an independent NMR sensor into the magnet control loop (even the very stable supercon magnets, on their own, are not stable enough; plus there are field variations of environmental origin). The jargon for this is NMR-lock (of the field).
Now that can be done by using a completely different "sample" than the one that is being measured - a separate, sealed ampule with its own coil and all. That is called "external lock" and yes, it is used sometimes, with its fixed "sample" containing nuclei like 2H, 1H, 19P, 11B, ...
However, the best stability in the measured sample area (the place where it matters) is achieved when the substance on which the lock is done is directly inside the sample (i.e., lock sample = measured sample) - that improves the stability by a factor of 3-5. Since in "normal" liquid NMR, the assayed substances need to be anyway dissolved in some solvent, it was natural to think about using the very solvent to lock the field. But in 1H-NMR, protonated solvents present two problems: (i) a 1H lock "channel" might (and does) interfere with the observed signals and (ii) strong protonated solvent signals cover the desired substance spectrum, especially at very low concentrations.
Hence the winning idea which neatly solved all these obstacles: use a deuterated solvent and operate the field lock at the deuterium Larmor frequency. It was so immensly successfull and is so commonplace today that even some present-day NMR spectroscopists no longer remember why they are doing it (kidding :-)).
Naturally, deuterium is also welcome because it is cheap: a liter of 99% D2O, for example, costs about 800$, so the cost per sample (say 0.2-0.5 ml, 2-4 bucks) is quite acceptable.
In proton NMR you don't want the solvent to dominate the spectrum so a solvent with deuterium replacing the hydrogens is a good idea.
The spectrometer also monitors the deuterium NMR signal and this allows frequency adjustment if drifting of the magnetic field (and frequncy synthesizer) takes place (particularly on long experiments). Though whether this is actually needed on modern super-conducting machines is debatable.
The deuterium frequency can also be used as a frequncy reference (so you don't have to add TMS).
Deuterium-labelled solvents such as CDCl3 (99.9% deuterium) are useful in NMR for a number of reasons. Primarily to prevent the NMR spectrum from being dominated by a very intense signal from 100%-CHCl3, which can produce unwanted baseline distortion, reduce integration accuacy and "hide" other NMR signals in the NMR spectrum. If 100%-CHCl3 was used then the parameters used to record the NMR spectrum would be optimised to this intense signal (receiver-gain), which would produce a poorer quaility NMR spectrum for the signals of interest. Using CDCl3 (99.9% deuterium) and a few miligrams of sample will ensure that the NMR parameters are optimised to the signals from the sample, and this will improve the signal-to-noise, provide a flat baseline and significantly reduce the chance of the residual solvent signal from hiding a signal of interest. The second most important reason for using CDCl3 rather than the cheaper CHCl3, is that the NMR-active spin 1 deuterium nuclei is used as a reference frequency to lock the NMR spectrometer to a signal a fixed and known frequency from the 1H NMR frequency of interest. This reduces the effect of magnetic field drift (which can be significant in older magnets), and provides a measure of absolute chemical shift reference. The deuterium signal is also used to improve the mangetic field homogeniety, "shimming" by using Z1, Z2, Z3 etc. in an itterative process prior to acquisition of the NMR spectrum. Modern NMR spectrometers map the magnetic field around the NMR tube, using the deuterium signal via a magnetic-resonance-imaging method.
It is possible to shim using the FID of the 100%-CHCl3 signal, though this is a method used by older NMR guys (like me).
The dominant signals in 100%-CHCl3 can be reduced using solvent supression (as in LC-NMR), or a relaxation filter.
We regularly use ordinary fully protonated solvents to record 31P NMR spectra to allow aliquots of reactions to be monitored.
There are three reasons why deuterated solvents are used in NMR spectroscopy.
Reason 1: To avoid swamping by the solvent signal.
There is usually much more solvent than sample in an NMR tube.
An ordinary proton-containing solvent would give a huge solvent absorption that would dominate the 1H-NMR spectrum.
Most 1H- NMR spectra are therefore recorded in a deuterated solvent, because deuterium atoms absorb at a completely different frequency.
But deuteration is never complete, so in CDCl3, for example, there is always some residual CHCl3.
You always get a solvent signal from CHCl3 at 7.26 ppm.
Reason 2: To stabilize the magnetic field strength.
The field strength of superconducting magnets tends to drift slowly.
Modern NMR spectrometers measure the deuterium absorption of the solvent and adjust the field strength to keep the resonance frequency (field strength) constant.
Reason 3: To accurately define 0 ppm.
The difference between the deuterium frequency and 0 ppm (TMS) is well known.
Modern spectrometers can "lock" onto the deuterium signal, so the addition of an internal reference like TMS is not usually required.