An example of a false negative is taking an HIV test and having the test come back negative to say the patient is clean, but in reality they have HIV. Another example of a false negative is a woman taking a pregnancy test saying "not pregnant" (i.e. test is negative), but she actually is pregnant. Between those two examples, it is better to have a false negative pregnancy test because it is non life threatening.
A false positive example would be getting bad news you have cancer, when you actually don't have cancer. Another false positive example is a test saying you have a cold virus, when in actuality you don't. The first example mentioned would have the patient likely go through intense chemo treatments which would greatly affect their livelihood. The second example is a more harmless false positive as it would involve at most a flu shot if anything.
