Here's how I look at it...
Sensitivity: for people w/ the disease, the liklihood of getting a positive test result.
Specificity: for people w/o the disease, the liklihood of getting a negative test result.
If you were designing a screening test, you'd want it to be as highly sensitive as possible, even if that meant picking up some false negatives.
For a confirmatory test (after you've identified all the positives by your highly sensitive test), you'd want it to be as highly specific as possible.
***Remember, sensitivity rules OUT, specificity rules IN.
For predicitive values...
Positive predictive value: for people w/ a positive test result, the liklihood of having the disease.
Negative predictive value: for people w/ a negative test result, the liklihood of not having the disease.
Predictive values are unique in that they depend on prevalence (the % of population w/ a dz at a given 'snapshot' in time). As prevalence rises, your PPV rises, and NPV decreases. That's because if a disease is highly prevalent, there's a good chance that a positive test result means that person has the dz!