Sensitivity and specificity in medicine is usually used in the context of testing as compared to the true result from the test and was first introduced in 1947.
The graphic from Wikipedia illustrates it well.
Sensitivity is the probability of a positive test result when the test is done on a true positive case
Specificity is the probability of a negative test result when the test is done on a true negative case
Usually there is a balance of sensitivity and specificity since if the test is made very sensitive it will start detecting false positives which get triggered because the test is very sensitive to small changes in the measured analyte. In medical contexts: A high sensitivity test can be useful for ruling out the disease since it will not miss the positive test.
On the other hand specificity is what makes it reliable meaning that it excludes tests that are truly negative. This becomes important because a non-specific tests leads to false result that would lead in the clinical environment, to expense, worry and more testing. In medical contexts: A high specificity test is useful for ruling in the disease since it does not give a false negative result.
Mathematically these terms are defined as
Sensitivity: number of true positives / (number of true positives + number of false negatives)
Specificity: number of true negatives / (number of true negatives + number of false positives)
