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Sensitivity and specificity mathematically describe the accuracy of a test which reports the presence or absence of a condition. Individuals for which the condition is satisfied are considered "positive" and those for which it is not are considered "negative".
If the true condition can not be known, a "gold standard test" is assumed to be correct. In a diagnostic test, sensitivity is a measure of how well a test can identify true positives and specificity is a measure of how well a test can identify true negatives. For all testing, both diagnostic and screening, there is usually a trade-off between sensitivity and specificity, such that higher sensitivities will mean lower specificities and vice versa.
If the goal of the test is to identify everyone who has a condition, the number of false negatives should be low, which requires high sensitivity. That is, people who have the condition should be highly likely to be identified as such by the test. This is especially important when the consequence of failing to treat the condition are serious and/or the treatment is very effective and has minimal side effects.
If the goal of the test is to accurately identify people who do not have the condition, the number of false positives should be very low, which requires a high specificity. That is, people who do not have the condition should be highly likely to be excluded by the test. This is especially important when people who are identified as having a condition may be subjected to more testing, expense, stigma, anxiety, etc.