Both “false-positive” and “false-negative” test-results exist.
1/ Positive and Negative tests-results
First, notice that the usual phrasing is “one is taking a test”, or “one is being tested for something”. One is “positive” for anomalies just like one is “positive” for alcohol, narcotics, drugs, disease, pregnancy, driving speed limit or amounts of goods at the customs: ie. typically some output values are above or below average. In everyday life, tests usually measure the concentration of a given chemical compound, or goods, or speeds; it’s the same for software: except that instead of chemical compounds or goods, we have file system objects, and we also have durations, speeds, loads, etc.
So, when software is taking a test, software is tested “for” regressions. Just like a chemical-test might reveal the presence of a molecule in your blood, a regression-test might reveal some incoherent or wrong behavior coming from the system-under-test. Therefore a “positive” test-result is a test-result that causes the test to appears, in first instance, and without further investigation, as “failed”.
By symmetry, a “negative” test-result is a result that did not detect any anomalies; the test is then usually considered as “passed”.
2/ False test-results
A “false-positive” test is a test that first appeared as “failed” but was later proven to be insignificant (eg. just like you would be first tested positive for something after saliva, or urine test, and later dismissed upon running more precise, blood tests).
Finally, “false-negative” tests are much more rare in practice: those are tests that did not catch an anomaly but should have. One can tell a test is a false-negative whenever one finds a problem that was not reported in the test covering the use-case.
In a nutshell:
Positive test-result = Anomaly detected = Failed test
Negative test-result = No anomaly detected = Passed test
False-positive test-result = Anomaly detected but no problem found after further investigation.
False-negative test-result = No anomaly detected and yet there was a bug.