Some degree titles have many components, e.g.:
Master of Arts in Psychology: Behavioral Health
|_A__|_B_____|_C___________|_D________________|
Another e.g.:
Master of Education in Curriculum and Instruction: Special Education
|_A__|_B__________|_C___________________________|_D________________|
What are these components called?
To take one of your examples:
Master of Arts in Psychology: Behavioral Health
|_A__|_B_____|_C___________|_D________________|
"Master" is the degree. Etymologically, "degree" comes from "degré": the step of a stair (or rung of a ladder). "Degree" can be synonymous with "level" or "extent" - it is a measure. You could say "the level of a degree", but that would be a tautology. It's not incorrect, but "level" is redundant there - "degree" already says everything that you'd be trying to get from "level". Degree is a very flexible word, and can be used to refer to just part A, parts A+B, parts A+B+C, and parts A+B+C+D.
Then either: "Psychology" is the subject and "Behavioral Health" is the speciality, or "Psychology: Behavioral Health" is the subject. You might also hear "field" used instead of "subject".
"of Arts" doesn't really have a well-recognised name. You could call it type, kind, field, sort, school or faculty, but in the end, if you want to ask what sort of Master's someone has, you'll probably end up saying something like: "What kind of Master's? Master of Science, Arts, MBA, something else?"
I would think "Master of Arts" is the degree, this is because in medieval universities the arts faculty was the first faculty one had to graduate from before being admitted to higher study. So the master of arts looks like it should be thought of as the name of a degree awarded by the arts faculty.
I am not aware of either (a) a consistent way of referring to the parts of the degree title or, more fundamentally, (b) a consistent way of deciding even what goes into the degree titles! None of my degrees have a colon or any listed specialty or sub-field. For that matter, what I translate from Latin as a masters degree is actually a Scientiæ Magister (which should properly be abbreviated SM, not MS).
According to academic tradition, all PhD are degrees in philosophy. As a result, a PhD in biology is a doctoral degree in philosophy (in the sense of thinking and research) with an specialty in biology. On the other hand, a DSci would be a doctoral degree in science in the field of biology. It's not clear to me that we would want to refer to the fields or specialties in that case the same way in any sort of rigorous way.
EnergyNumbers' answer seems like a good attempt to parse out the answer. That said, I agree with Shane's answer that "Master of Arts" is the name of the degree. So, in:
Master of Arts in Psychology: Behavioral Health
|_A__|_B_____|_C___________|_D________________|
I would say that A+B is the degree. A is not the degree because an MBA is simply not the same degree as an MS, MFA, MPH, MPhil, etc.
C is the field, subject, area, or specialty. D is the sub-field, sub-area, or sub-specialty. I think adding a "sub-" makes this relationship clear.
I would call "A" the type, or maybe the level, of the degree. I like level because if I were to ask someone what their highest level of degree was, I would expect to hear something like doctorate, masters, bachelors, or associates and would not expect to hear that it was a bachelors of science.
