To approve ordinarily means to regard favorably, but its biblical and archaic sense is stronger: to test, prove, and accept as genuine after examination. The related idea is that what withstands testing is approved, while what fails is rejected or reprobate.
The New Testament speaks of persons approved in Christ, believers who are approved, presenting oneself approved to God, and those approved after testing. The word is also used of proving what is acceptable and of tested faithfulness.
The underlying term was commonly used for testing ores. Its meaning includes both the examination and the favorable verdict that follows when the object stands the test.