Hey Jaymes,
Karl Popper's major contribution to the philosophy of science was his attempted solution of the demarcation problem, which is the problem of how we distinguish science from non-science. Popper was concerned that many putative scientific theories like Marxist interpretations of history, psychoanalysis or Adler's individual psychology were not fully testable because the observational consequences of a particular test could always be interpreted to be in favor of the theory. For example, if a Freudian believes that you have an unconscious drive to have sex with your Mother and decided to put this to the test by asking you, you can only respond in two general ways: "yes, I have thoughts of having sex with my Mother" and "no, I don't have thoughts of having sex with my Mother". However, Freudians can count either result as being in favor of their theory because it's an
unconscious drive which you're not supposed to be aware of.
So Popper's solution was to say that in order for a scientific theory to count as scientific the theory in question must be falsifiable by some observations. To clarify a bit, Popper uses the terms testable and falsifiable interchangeably because for Popper a test of a theory doesn't count as a real test unless there's the possibility of the theory being shown to be false.
As for your question about whether or not the existence of mental disorders is testable and falsifiable, I think that's a tough question. Psychologists and psychiatrists are heavily influenced by a position in the philosophy of science known as operationalism which defines the concepts used in science (i.e. time, weight, intelligence, etcetera) in terms of the operations in which they are measured. So on the operationalist's account, there's no distinction between mental disorders and the way that we measure them. For example, Borderline Personality is when someone meets at least 5 out of 9 of the critiera put forward in the DSM-TR-4. You can't ask "yeah, but what is it really?" because that's all there is to the operationalist story.
Operationalists pretty much do science this way: they come up with a concept, explain the concepts grammar, develop a metric for measuring the concept, validate the metric by looking at what the scores correlate and predict, then build and test theories about the concept.
So I guess, to answer your question, the existence of mental disorders isn't really falsifiable and wouldn't pass Popper's test, but most psychologists and psychiatrists, who aren't just totally unreflective about theoretical questions, see the the existence of mental disorders as more of question about what we mean by the concepts that we use and how we decide to measure them.
Of course, there are lots of different psychologists and psychiatrists who believe that psychological predicates are floating around in our heads and the main problem is that we just haven't found them yet, but I think this is just too confused to take seriously. The operationalists seem to have a much more coherent philosophy of science.