The most common technical term would probably be "spinal manipulation" in general, and "manipulation of the cervical spine" / "cervical spinal manipulation" to denote the treatment of the neck portion of the spine specifically.
"Cavitation" is the term for the sound that can be produced during the quick separation of joint surfaces.
Unfortunately, @Desai seems to be not quite correct to include the term "crepitation" or "crepitus" here, however, it is related to the sounds that joints can produce. My understanding is that crepitus is frequently caused by pathologic conditions (for example hypermobility, damage of the joint cartilage, muscle weakness) and rather not as a common side effect during spinal manipulation.
The technique that brought you to ask your question here is a general technique. There are general techniques for "hitting several joints at once" (I observed that a certain therapist would rather be hoping that the pain-causing segment would be affected by the general technique) - and specifically targeted techniques which require more skill and experience for adequate localization and angular position of the joint (portion) to be treated, as well as for the execution of the technique itself. But these specific techniques also require more time, as well in practicing as also in execution... chiropractors, orthopedic manual therapists or doctors of osteopathic medicine may have the greatest expertise with "High-Velocity-Low-Amplitude"-techniques, as "spinal manipulation" is sometimes also called ("HVLA"-techniques).
"Low amplitude" suggests that "if you use high velocity" (apply more force than with the softer, slower "mobilisation", which is probably the central term in manual therapy), then "you better head for a very small movement that you try to guide the joint to". If you think of what these words mean, it could be easy to "go a little bit too far". It is much easier for a therapist - and safer for the patient - to use (slower) mobilisation instead of (faster) manipulation techniques. With them, you have a lot more time, during the application, to change direction, position, intensity, duration ... so, your own brain has more time to actually learn "how joints feel". As a learning professional, I do find value in such a viewpoint.
While the evidence base supports the use of these techniques in modern musculoskeletal practice (1), the effect of manipulation techniques is correlating with the kind of subjective hope (expectation) patients may direct towards it. For example, the source I use here states that the technique of spinal manipulation may cause "a reduction of [perception] of pain" (1).
The original goal of reflexively applied manipulation and mobilisation techniques is, of course, to improve range of movement exactly where it is needed (nowhere else, hopefully). There is a somewhat clarifying story worth to tell, written by one of the more famous manual therapists, Geoffrey Maitland. In the 3rd edition of his book about spinal manipulation (2) he reflects that ["a decade ago, of the patients that would improve on application of passive moment, 85% would need mobilisation, while 15% would require manipulation"]. One sentence later, he displays his personal professional development in stating that he came to the conclusion that a rate of 99% (mobilisation) to 1% (manipulation) seemed to be the more adequate estimate for him.
While that would have been a good point to end this little trip through the universe of spinal manipulation, it is still worth to at least mention the bias or blindness that generalized perception seems to have towards specific mobilisation, which is less effect-presenting, but not less effective. Also, while both methods are known to not permanently relieve pain, I feel the serious obligation to argue that adequate mild heel lifts - even as a back pain prevention tool for the general population - will push overall symmetric movement in all our spines facet joints on the long run (and thus lower back pain efficiently). For the biomechanical genesis of spinal pain, manual therapy and its relatives provide diagnostic skill that, if taken seriously, by far transcends its "special fx unit" :) ... for example, functional radiology is not up to this level of restricted movement detection yet, it is, by today, very much still a static radiology facing very specific and localizable problems from its own and evolved angle. When we speak to it about movement ... the current focus of therapies is upon exercise and load transfer (to my mind, mild leg length differences are gravely overlooked in modern clinical practice, according to the high influence of their causal link to spinal pain, even enough to with some urgency refer towards this phenomenon in an article about spinal manipulation).
(1) Grieve's Modern Musculoskeletal Therapy, 4th ed., 2015, p. 277 ff.
(2) Maitland, Manipulation der Wirbelsäule, 3rd ed., 2006, p. 182