In modern translations this phrase is usually rendered “shows no partiality” (NRSV) or “shows no favouritism” (NIV). The context is Peter’s visit to the Gentile, Cornelius, a Centurion of the Italian Regiment. To grasp the full force of a word or phrase it often helps to go back to its literal meaning and work from there.
The Greek word for the phrases quoted above is prosopoleptes. This is a compound of two words, prosopon = ‘face’, and lempsis = ‘a receiving’. Literally, the word means ‘one who receives another according to his face’ – i.e. according to his appearance and the judgements as to wealth, status and position that human beings make about others from their appearance. Peter has learnt that God does not judge people by human standards but by His own, and is therefore “no respecter of persons”. See also John 2:24-25.