Tony Chan is right about "covered", but I wanted to provide some additional background, from the Bible Knowledge Commentary, as we can see what this cover is:
It cannot be unequivocally asserted but the preponderance of evidence
points toward the public head covering of women as a universal
custom in the first century in both Jewish culture ([apocryphal] 3
Maccabees 4:6; Mishnah, Ketuboth 7. 6; Babylonian Talmud, Ketuboth
72a-b) and Greco-Roman culture (Plutarch Moralia 3. 232c; 4. 267b;
Apuleius The Golden Ass 11. 10). The nature of the covering varied
considerably (Ovid The Art of Love 3:135–65), but it was commonly a
portion of the outer garment drawn up over the head like a hood. It
seems that the Corinthian slogan, “everything is permissible,” had
been applied to meetings of the church as well, and the Corinthian
women had expressed that principle by throwing off their
distinguishing dress. More importantly they seem to have rejected the
concept of subordination within the church (and perhaps in society)
and with it any cultural symbol (e.g., a head-covering) which might
have been attached to it. According to Paul, for a woman to throw off
the covering was an act not of liberation but of degradation. She
might as well shave her head, a sign of disgrace (Aristophanes
Thesmophoriazysae 837). In doing so, she dishonors herself and her
spiritual head, the man. 11:7–9. The man, on the other hand, was not
to have his head covered because he was the image and glory of God.
Paul based this conclusion on Genesis 1:26–27. A woman’s (a wife’s)
glory and image was derived from (1 Cor. 11:8) and complementary to
(v. 9) that of the man (her husband). Man, then, was God’s
authoritative representative who found in woman a divinely made ally
in fulfilling this role (Gen. 2:18–24). In this sense she as a wife is
the glory of man, her husband. If a married woman abandoned this
complementary role, she also abandoned her glory, and for Paul an
uncovered woman’s head gave symbolic expression to that spirit.
Lowery, D. K. (1985). 1 Corinthians. In J. F. Walvoord & R. B. Zuck (Eds.), The Bible Knowledge Commentary: An Exposition of the Scriptures (Vol. 2, p. 529). Wheaton, IL: Victor Books.
This is again why we must keep in mind that although these letters are part of the cannon, we cannot lift sentences out of them and apply them directly to our lives or compare them to other sentences lifted out of context from other passages. We have to understand the context: who was the letter written to, what issues was it addressing in that particular community, and then what is the spiritual message for us in our own (very different) communities.
This requires wisdom and proper hermeneutic principles (e.g. require multiple witnesses in scripture). In this case, Paul was responding to an issue of Church discipline in the Corinthian church regarding the comportment of some of the women in choosing a dress style that would offend others in the community and shame their husbands.
So the real underlying principle from which these exhortations spring is that there is no room in the gospel for revolutionary social movements as we are not to mind the things of the world at all. If we live in communities where something is considered shameful, we are not to violate those social conventions because we have newfound freedom in Christ.
Just as Paul said in his letter to the Galatians: Gal 3.27-28
For all of you who were baptized into Christ have clothed yourselves
with Christ. There is neither Jew nor Greek, slave nor free, male nor
female, for you are all one in Christ Jesus.
Yet Paul also said in his letter to Corinth (1 Cor 9)
For though I am free from all, I have made myself a servant to all,
that I might win more of them. To the Jews I became as a Jew, in order
to win Jews. To those under the law I became as one under the law
(though not being myself under the law) that I might win those under
the law. To those outside the law I became as one outside the law (not
being outside the law of God but under the law of Christ) that I might
win those outside the law. To the weak I became weak, that I might win
the weak. I have become all things to all people, that by all means I
might save some.
Is that a contradiction? How could Paul give different advice to different churches? We have to understand the unifying principle to see why in some cases one is the proper exhortation and in another case a different exhortation is needed. The unifying theme is we have spiritual freedom in Christ but in the world we give up all freedom and rights, because we are not to mind of the things of the world at all:
“You have heard that it was said, ‘Eye for eye, and tooth for tooth.’
But I tell you, do not resist an evil person. If anyone slaps you on
the right cheek, turn to them the other cheek also. And if anyone
wants to sue you and take your shirt, hand over your coat as well. If
anyone forces you to go one mile, go with them two miles. Give to the
one who asks you, and do not turn away from the one who wants to
borrow from you. (Matt 5.39-42)
And in Paul's letter to the Phillipians:
Brothers, join in imitating me, and keep your eyes on those who walk
according to the example you have in us. For many, of whom I have
often told you and now tell you even with tears, walk as enemies of
the cross of Christ. Their end is destruction, their god is their
belly, and they glory in their shame, with minds set on earthly
things. But our citizenship is in heaven, and from it we await a
Savior, the Lord Jesus Christ,
So the spiritual freedom applies to the spiritual man whose mind is set on spiritual things and his walk is in heaven, and to the carnally minded man whose mind is set on the things of this world, the only message is the cross -- lay down your life. Then as your mind shifts from the things of the world to things in heaven, you have freedom and unity and all things are yours. But to use the rationale of spiritual freedom to try to demand worldly rights is an abomination worthy of rebuke, and this is the rebuke that Paul delivers here to the women of Corinth even as he has a very different exhortation to the Church in Galatia and a warning to the Church in Phillipi.