Convergent validity is whether a test correlates strongly with other established measures of the same or a closely related construct.
If a new, shorter extraversion questionnaire is any good, people who score high on it should also tend to score high on longer, well-established extraversion measures, and their answers should line up with things extraversion is known to relate to, like how much they enjoy parties or speak up in group settings. That overlap is convergent validity: independent measures of the same underlying thing "converging" on similar answers, even though they're worded differently and were built by different people.
It's one half of a pair. Discriminant validity is the other, checking that a test does not correlate with things it shouldn't. Together, strong convergence with related measures and weak correlation with unrelated ones make the strongest case that a test is measuring a real, distinct construct rather than something vague or overly broad. Neither one alone is convincing — a test that correlates with everything isn't showing convergent validity so much as a lack of discriminant validity in disguise.
Convergent validity is one of the more common checks researchers run early when validating a new personality measure, since comparing it against an already-trusted questionnaire is faster than waiting on real-world outcomes, which is what criterion validity requires instead.