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Convergent validity demonstrates that measures of the same construct correlate substantially with each other. Discriminant validity shows that measures of theoretically distinct constructs do not highly correlate. Together, they establish the distinctiveness and appropriateness of a construct's conceptualization.
From your work with construct validity and factor analysis, you know that a psychological construct like "anxiety" or "conscientiousness" is a theoretical entity — it does not exist in the world the way weight or temperature do. To argue that a test actually measures the construct it claims to measure, you need a body of evidence showing how scores relate to other measures. Convergent and discriminant validity are the two sides of that evidence: one shows that your measure *agrees* with other measures of the same thing; the other shows that it *disagrees* with measures of different things. You need both.
Convergent validity is demonstrated when scores on your measure correlate substantially with scores on other instruments that are theoretically supposed to tap the same construct. If you have developed a new measure of depression, it should correlate strongly with the Beck Depression Inventory and the PHQ-9. If it does not, one of two things is wrong: either your measure isn't capturing depression, or the existing measures aren't either. High convergent correlations provide evidence that multiple independent operationalizations are converging on the same underlying reality. The logic is triangulation — if different methods (self-report, behavioral observation, physiological measure) all point to the same construct, confidence grows that the construct is real and that each measure is capturing it.
Discriminant validity is demonstrated when your measure does *not* correlate highly with measures of theoretically distinct constructs. Your depression measure should correlate moderately with anxiety (the constructs are related but distinct) but should not correlate as highly with extraversion or intelligence. If two supposedly distinct constructs correlate near 1.0, they are empirically indistinguishable — which means either the constructs are the same thing, or the measures are so blunt that they cannot separate them. Discriminant validity failures often reveal method variance: two self-report measures will correlate partly because they share the method, not because they share a construct. This is why Campbell and Fiske's multitrait-multimethod matrix (MTMM) — which your factor analysis background prepares you to interpret — examines convergent and discriminant patterns across multiple constructs measured by multiple methods simultaneously.
The practical implication is that both forms of validity evidence are necessary and neither is sufficient alone. A measure that converges with everything (including unrelated constructs) demonstrates only that it captures something broad, perhaps acquiescence or social desirability. A measure that discriminates sharply from everything (including theoretically related constructs) may be too narrow or poorly operationalized. The sweet spot — strong convergence with the same construct, modest correlation with related constructs, and low correlation with unrelated ones — is what distinguishes a well-validated instrument from one that merely looks face-valid.