Personality type really boils down to simple matrices of behavioral preferences.
The two main western strains are:
"Classic temperament", which is based on a matrix of how a person tends to express to or "approach" others in interactions (this gives us the classic "introversion" and "extroversion"), and also how they "respond", or how much they want others to approach them.
This yielded the ancient "four temperaments", of Hippocrates and Galen, which were originally believed to tie to body fluids ("the humors"), which were tied to the "elements" (earth, air, fire, water) via them being either "hot or cold" (expressive), or "moist or dry" (responsive).
By the 17th century, it was realized that body fluids had nothing to do with them, but they were still good descriptors of our interactions.
The other main system is MBTI, based loosely on Jung's theory, which starts with introversion and extraversion, and then adds a dichotomy for perception (Sensing vs intuition [N], or more practical or hypothetical awareness), one for "judgment" (Thinking vs Feeling, which is whether we are more "impersonal", or consider people more in our decisions), and then Judging or Perceiving, which involves which of those last two we tend to "use on the outside world" (as opposed to keeping them inside our own reflection), which was deemed important in interpersonal interaction. It overall covers stuff like whether we are are "open" to new information, or desire more "closure".
This produces sixteen types based on the possible combinations of those four dichotomies.
I have said this is reflecting how we "divide reality", even comparing it to the actual dimensions of space and time (we look one direction, and the opposite direction is still there, but not "conscious" to us, until we turn that way. The parallel directions we can see partly in).
Encountering both systems years ago, I determined to see how exactly they fit together, and while I/E is pretty much the same in both, representing expressiveness; T/F and J/P loosely correspond to the other factor of responsiveness. S/N is something held in common by opposite temperaments.
The classic temperaments were actually repopularized by familiar Christian writer Tim Lahaye. Another Christian ministry, the National Christian Counselor's Association, is led by a Methodist couple, the Dr. Richard and Phyllis Arno, in Florida. This was the system my wife encountered and was licensed in for awhile, exposing me to type theory.
It licensed a once familiar instrument called the FIRO-B, which took the expressive and responsive matrix and added ten point scales to it, so that there were moderate ranges. From this, the Arno's discovered a fifth temperament. It also created three levels of temperament: social, leadership, and deeper relationships, explaining the "blended" temperaments LaHaye and others used.
Personality theories of course had their origins in non-Christian sources, like ancient pagans, and others (and especially with the connection to the elements at the center of astrology), and before LaHaye, the classic temperaments were used by Rudolf Steiner who was rather mystical.
You did have Jewish rabbi and physician Maimonides, who was a step in their development, though.
Jung of course, was into alchemy, which he often used in his studies of dreams and interpretations.
So it's understandable that many Christians might think these theories are bad, as they had been critical of all secular psychology, for supposedly teaching "secular humanism". So in IFB circles and some Reformed and other deeply conservative movements, it's often gone right along with anti-CCM, KJVO-ism and the rest of "separationism". (Bobgans, Biblical Discernment Ministries, David Cloud, etc. who will criticize ministries as conservative as CRI or one of the Bob Joneses for using just a term associated with psychology. And even LaHaye and Arno omit any mention of Jung in their books).
But psychology is just "study [-logy] of the mind [psych-]. People think any discussion of the mind that doesn't attribute all of our behavior to "sin" is therefore "excusing" it and "compromising" the Gospel. The Gospel is the message of how we had fallen into sin, and God's means and offer of forgiveness for it. Identifying patterns of behavior that may play out in our interactions (sinful or not) is not contradicting this in the least. (Unless you take this exaggerated position that there is absolutely "no time for discussion of anything else", except for getting yourself and others saved. That would render all non-evangelistic activities and sicussions sinful).
Now, with all of that said, Enneagram is another system, that has become very popular in type discussions, and even unofficially added to MBTI type to create more variations. In my view, they resemble the five temperaments plus blends, but it's an entirely different framework from the other systems, based on triangular matrices (hence, 3×3=9).
One of it's leaders, Gurdjieff, is a known mystic, and what struck me was the main figure used in the system, which is an actual geometric "enneagram" (9-pointed "star" polygon), but with two of its comprising triangles broken open, so that their lines meet at types 4 and 5, which is supposed to convey some sort of connection between different types. But it looked to me like some sort of mandala, so I've always been a bit more leery of that system. (Never seen or heard of anyone using it that way, though the way you'll see it displayed on a site just looks funny). Some claim the theory is more about are "reactions", and that it may not be a real inborn "type". So again, it's been stuck on at the end of people's MBTI types.
But in any case, all of this is just categorization of people according to cognitive preferences that do shape our behavior. Like when I think back to the most vigorous debates I've had here, I figure those people were likely NTJ types. NT's are the "Rational" types, who like to argue abstract concepts like theology and politics (which the S types tell us should never be argued, because practically, they can't ever be resolved and just cause endless conflict).
The larger secular science (including "mainstream" psychology) rejects it, pretty much the same way they reject Creationism, because of its lack of "empirical evidence", at least that measures up to their standards. Both typology and religion often end up "proven" anecdotally ("it works for me; this is my experience", etc.), which is technically empirical, but they want big formal "studies" that the "peers" all approve of.
MBTI does have a lot of statistical work behind it, but these still generally aren't accepted. You even have a few theorists doing neurological studies (such as brain scan work), to try to directly connect it to the elements of type, but this still isn't impressing the higher ups (as most Christians would testify, from the creation debate, science can be pretty biased).
As it stands now, the personality system that is most respected "scientifically" is something called the Five Factor Model or "Big Five" (with the main test being the NEO-PI), whose five factors are similar to to MBTI, with the addition of one called "Neuroticism", which was deemed as too "negative" by Myers.
This isn't a "type" system; it is just five standalone behavior "trait" scales that are not factored together into "types" (though there are some imitation systems that do that, and one that adds what they call "Assertive/Turbulent" to MBTI type). It seems that sort of thing is distrusted (too "amateurish"/parlor-gamish", or even "astrology"-like, which is the common criticism for them?) in the field. But a lot of people does find it makes sense for them.