3 Paul does not use a word specifying what type of “covering” is meant (veil, hat, etc.). The Greek word he uses here (ἐξουσία exousia; translated symbol of authority) could be (1) a figure of speech that may substitute the result (the right to participate in worship) for the appropriate appearance that makes it possible (the covered head). Or (2) it refers to the outward symbol (having the head covered) as representing the inward attitude the woman is to possess (deference to male leadership in the church).
4 Paul does not explain this reference to the angels, and its point is not entirely clear. It seems to reflect an awareness that angels are witnesses to church life (cf. Eph 3:10) and would be particularly sensitive to resistance against God’s created order.

In other versions

That is why a wife ought to have a symbol of authority on her head, because of the angels.

It is for this reason that a woman ought to have authority over her own head, because of the angels.

This is why a woman should have a symbol of authority on her head, because of the angels.

For this reason, and because the angels are watching, a woman should wear a covering on her head to show she is under authority.

Because of this, the woman ought to have a symbol of authority on her head, on account of the angels.

For this cause ought the woman to have power on her head because of the angels.

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