As cultural animals, humans have come up with many ways to describe the ways that particular individuals tend to behave, and also have devised many ways of categorizing individuals (personality types).
Since the social world is often the most important aspect of the human environment, from an evolutionary perspective, it is not surprising that people are highly motivated to understand any regularities in the social environment that could be exploited, just as we are interested in useful properties of the material environment (what is good to eat, and so on.) This is a sort of theory building (Story), which is useful descriptively (in communicating our understandings about people) and predictively (in anticipating how they tend to behave.)
Given that people do have behavioral regularities, this must somehow be manifested in the physical structure and connectivity of their brain, which we would also expect to be under strong generic influence. That is, personality is real not only in the sense that it is useful, but also in having identifiable relations to measurable quantities. See heritability of the big five.
If significant variation persists in a population, then an evolutionary approach primes us to look for situational advantages of those differences. If there was one best way to be (most Adaptive Behavior), then we would expect everyone to converge to that ideal. The persistence of behavioral differences allows individuals to exploit different behavioral niches (different strategies.)
See also Personality Psychology and The Nature of Psychological Maturity. The reality of personality and its significant hereditary component is also inconsistent with philosophical conceptions of free will as an uncaused cause.
Whenever someone says “there's two kinds of people…” they are proposing a personality type. This has presumably been going on since prehistory, and early recorded theories of personality take the form of types (see the Four temperaments). Attempts to quantify personality using rating scales and statistical analysis are a modern elaboration that is more rigorous, but does not supersede personality typing.
Formally, the obvious difference between a type and a trait is that a type is discrete, whereas a trait is continuous. For example, in the Myers-Briggs Type Indicator, a person is either introverted or extroverted, whereas in the Big Five personality traits, extroversion can be an in-between value.
There is a more subtle yet far more important difference. The quantitative trait approach allows us to say in a rigorous way that, of the hundreds or thousands of personality characteristics, certain characteristics go together. For example, this research tells us that if a person is friendly and likes to be with people, then he is also rather likely to be cheerful and assertive. That is, trait research is oriented toward finding general regularities across the entire population. The correlation between these more specific traits suggests that there is some underlying common cause in the brain, and this interpretation is supported by the heritability of traits.
While personality typing does imply that traits go together, typing is particularly valuable as a way to appreciate the specifics of an individual. A person is far more than the sum of their parts; particular combinations of personality traits are meaningful in themselves because they predict certain sorts of behavior patterns and aptitudes, both in intimate relationships and also broader social contexts. That is, personality types describe the social relevance of particular kinds of personalities. For example, a discussion of a personality typology such as Myers-Briggs will often mention typical occupations that are chosen by or are suitable for a particular type, and personality typing is also frequently proposed as a way for psychologically savvy managers to appreciate how to best motivate and direct their employees.
Although quite a few personality type systems have tests that can be used to assign a personality type, outside of the management context we can't ask people to take a test so that we can better manipulate them, so the practical application of personality type theories is usually based on the user's subjective impression of a person's type.
The big five is a quantitative (trait) personality system. This approach dominates personality psychology research because it has a sound statistical basis and is the result of a convergence of many different-seeming trait schemes. Personality type tests are created top-down by taking the already formed type theory and then designing questions that “get at” the desired personality dimensions. In contrast, big five personality took a bottom-up approach, asking people to rate how well they were described each of the words on a list of personality-related adjectives, then finding which words hung together.
One of the quirks of this approach is that Factor Analysis gives a statistically rigorous way to define “hanging together”, but it doesn't tell us what to call the dimensions it identifies. The names below are generally used in research literature, but there is plenty of room for disagreement about whether those names best capture the implied category.