For this discussion, I will be utilizing several Bible passages. I will go ahead and produce them below:
I desire then that in every place the men should pray, lifting holy hands without anger or quarreling; likewise also that women should adorn themselves in respectable apparel, with modesty and self-control, not with braided hair and gold or pearls or costly attire, but with what is proper for women who profess godliness—with good works. Let a woman learn quietly with all submissiveness. I do not permit a woman to teach or to exercise authority over a man; rather, she is to remain quiet. For Adam was formed first, then Eve; and Adam was not deceived, but the woman was deceived and became a transgressor. Yet she will be saved through childbearing—if they continue in faith and love and holiness, with self-control. (1 Timothy 2:8-15; ESV)
Women should "adorn themselves in respectable apparel, with modesty and self-control, not with braided hair and gold or pearls or costly attire, but with what is proper for women who profess godliness—with good works." What does this look like? What would a respectable, modest, and self-controlled woman look like? The reality is that modesty is a bit more than just looks. I will simply address looks first and then work through the other aspects of modesty. A modest woman is one who does not leave room for men to be tempted into sin by dressing revealingly. A modest woman is someone who does not dress to bring attention to herself. Modesty can be achieved by a woman concealing her body shape with loose fitting clothes. A modest woman will not manipulate her signs of age in order to be found more visually appealing to men. For example, a modest woman should not dye her hair, wear excessive makeup, or commit to cosmetic surgeries.
"Neither is the hair to be dyed, nor gray hair to have it’s color changed [...] Old age, which conciliates trust, is not to be concealed." - Clement of Alexandria
Yet these things also apply to men. Men likewise should be careful not to dress immodestly. A modest man will wear loose-fitting clothes and not clothes that reveal the shape of the body. Modest men will not dress so as to draw attention to themselves. Modest men will not try to conceal their age with dyed hair or cosmetic surgery.
You will also note that the passage says women should not have "braided hair and gold or pearls or costly attire." I believe this is more than likely in reference to fashion trends at the time Paul wrote this letter and not necessarily a hard and fast rule for women of all eras. But that does not mean we cannot extract any meaning from this section. What we can gather from it is that women should not attempt to display their wealth in the form of jewelry or other designer apparel. I believe that the this can and should be applied to men as well. Men should not wear their wealth in the forms of jewelry. An example of something like this would be a Rolex wristwatch.
Regarding the braided hair, I believe that this is in reference to time-consuming fixations with hair. Or even costly and timely hair procedures at a salon. Women should not spend too much time trying to fix their hair and neither should men. We, both men and women, should not become consumed with the tendency to try and make our hair perfect. Do we not have more valuable things to spend our time on?
"If it is true, as it is, that in men, for the sake of women (just as in women for the sake of men), there is implanted, by a defect of nature, the will to please; and if this sex of ours acknowledges to itself deceptive trickeries of form peculiarly its own—such as to cut the beard too sharply; to pluck it out here and there; to shave round about the mouth; to arrange the hair, and disguise its hoariness by dyes; to remove all the incipient down all over the body; to fix each particular hair in its place with some womanly pigment [...] all these things are rejected as frivolous, as hostile to modesty." - Tertullian
Modesty is not limited to just physical expressions (i.e., clothing, hair, makeup, etc.) but also mindsets and practices. I briefly touched on an immodest mindset with the example of time spent on hair. It is immodest to be consumed with the image of yourself. Whether this is hair, clothing, or something else. Our minds should not be overly occupied with our physical appearance, as our mind is to be occupied with the Lord. We are supposed to be humble, as Christ was. Christ is the perfect example of humility. He did not occupy His time considering Himself. Rather, He occupied His time with the problems of others. We should aim to be no different.
Nor is modesty limited to only mindsets regarding physical expressions. Paul goes on to say that a woman is to "learn quietly with all submissiveness. A woman is not permitted to "teach or to exercise authority over a man; rather, she is to remain quiet." This is the way of a modest woman. A modest woman knows her role in the hierarchy of social order. An immodest woman would seek to undermine this order that is established by God in order to advance herself. In other words, an immodest woman is not humble.
Or do you not know that your body is a temple of the Holy Spirit within you, whom you have from God? You are not your own, for you were bought with a price. So glorify God in your body. (1 Corinthians 6:19-20; ESV)
This passage applies to both men and women. If a Christian man or woman dresses so as to bring attention to themselves for ungodly reasons then they are not glorifying God in their body.
Now I commend you because you remember me in everything and maintain the traditions even as I delivered them to you. But I want you to understand that the head of every man is Christ, the head of a wife is her husband, and the head of Christ is God. Every man who prays or prophesies with his head covered dishonors his head, but every wife who prays or prophesies with her head uncovered dishonors her head, since it is the same as if her head were shaven. For if a wife will not cover her head, then she should cut her hair short. But since it is disgraceful for a wife to cut off her hair or shave her head, let her cover her head. For a man ought not to cover his head, since he is the image and glory of God, but woman is the glory of man. For man was not made from woman, but woman from man. Neither was man created for woman, but woman for man. That is why a wife ought to have a symbol of authority on her head, because of the angels. Nevertheless, in the Lord woman is not independent of man nor man of woman; for as woman was made from man, so man is now born of woman. And all things are from God. Judge for yourselves: is it proper for a wife to pray to God with her head uncovered? Does not nature itself teach you that if a man wears long hair it is a disgrace for him, but if a woman has long hair, it is her glory? For her hair is given to her for a covering. If anyone is inclined to be contentious, we have no such practice, nor do the churches of God. (1 Corinthians 11:2-16; ESV)
Again, we see the Scriptures reiterate the hierarchy of man being the head of woman. It is made plainly obvious through this passage that in Corinth they had a cultural tradition of women covering their heads to show submissiveness to the men. If a woman rejects this sign of submissiveness for the sake of self-gain, she should shave her head as it would be equally as shameful. Woman was created for man and it was not the other way around. Hence, she is to be submissive. However, we should not neglect the fact that men also have responsibilities in the relationship. Though I do not believe those responsibilities necessarily fall into the discussion of modesty. Modest women will respect their position as below men and submissive to men. They will remain humble and not seek to disturb the God-given order of things.
Regarding what Paul said about hair length, men should not grow long hair as there is no purpose in it. Women keep long hair to cover the pleasantness of their faces and the beauty that draws attention to them. This is not the case for men.
Likewise, wives, be subject to your own husbands, so that even if some do not obey the word, they may be won without a word by the conduct of their wives, when they see your respectful and pure conduct. Do not let your adorning be external—the braiding of hair and the putting on of gold jewelry, or the clothing you wear—but let your adorning be the hidden person of the heart with the imperishable beauty of a gentle and quiet spirit, which in God's sight is very precious. For this is how the holy women who hoped in God used to adorn themselves, by submitting to their own husbands, as Sarah obeyed Abraham, calling him lord. And you are her children, if you do good and do not fear anything that is frightening.
Likewise, husbands, live with your wives in an understanding way, showing honor to the woman as the weaker vessel, since they are heirs with you of the grace of life, so that your prayers may not be hindered. (1 Peter 3:1-7; ESV)
And, again, we are told of the hierarchy. Women are to be subject to men. Though it is in this passage that we see the true intention and reason behind modesty:
Do not let your adorning be external—the braiding of hair and the putting on of gold jewelry, or the clothing you wear—but let your adorning be the hidden person of the heart with the imperishable beauty of a gentle and quiet spirit, which in God's sight is very precious.
Women are supposed to be humble in their relation to men. They are to respect man's God-given authority over them. Regarding physical modesty, the woman is to do this because the beauty of her spirit is so much more precious than that of external beauty. Submitting to a man and humbling oneself becomes easier when the man came to you for the preciousness of your spirit and not for the external signs that disappear in the grave. The external signs fade with age. And if the man came to you to honor your external shell, then how can that honor last?