Discriminant validity is whether a test avoids correlating strongly with measures of constructs it shouldn't be related to, showing it measures something distinct.
Take Conscientiousness again: it should correlate with things like punctuality and follow-through, but it shouldn't correlate meaningfully with something unrelated, like shoe size. If a new conscientiousness scale turned out to correlate strongly with shoe size, or with an entirely different trait like extraversion, that would be a red flag — a sign the test is picking up something other than the specific construct it claims to measure, whether that's a flawed item set or unwanted overlap with a neighboring trait.
Discriminant validity is the necessary counterpart to convergent validity. Convergent validity shows a test lines up with things it should relate to; discriminant validity shows it doesn't blur into things it shouldn't. A test with strong convergent validity but no discriminant validity usually means the test is too broad, correlating with almost anything you compare it to rather than picking out something specific.
This pairing is also central to the idea of a nomological network: a real construct has a lawlike pattern of what it does and doesn't relate to, and discriminant validity is essentially the "doesn't" half of that pattern, checked deliberately rather than left to chance.