to look into something closely in order to size it up: to investigate, question, or weigh it toward a verdict
At its core the word names a careful sifting: you look into a matter closely, turn it over, and form an assessment. The same act shows up in three settings across the New Testament. In a courtroom it is the questioning that establishes the facts, as when Pilate examines Jesus and reports that he found no charge that would stick. In the study it is patient investigation, the daily searching of the Bereans who checked Paul's preaching against the Scriptures. In Paul's letters it becomes the weighing of a person or a spiritual truth, sizing it up to see what it really is.
What holds these uses together is that the word describes the process of examination, not the finished verdict. This is why Paul can set it against the verb for reaching a decision, κρίνω: he is willing to be investigated by the Corinthians, and even to have his own conscience probed, because being examined is not the same as being sentenced. The final ruling belongs to the Lord alone.
The spiritual uses in First Corinthians press this further. Paul says the things of God's Spirit can only be appraised by someone the Spirit has taught, and that such a person can weigh everything while being fully understood by no one. Here the word comes close to our idea of discernment: a trained, penetrating assessment rather than a quick opinion.