This is the approved revision of this page, as well as being the most recent.
There is more about modesty than dress. Primarily modesty relates to a person's conduct privately, publicly, and personal self-awareness. 
As a Middah
- Hashem displays his modesty by not openly revealing His awesome presence and to us appears hidden. We are enjoined to follow in His ways (which is the mitzvah of VeHalachta BeDerachav) and do the same by not looking for public attention in a general sense.
- A Sephardi woman, according to Rav Ovadia, should cover her arm down to and including the elbow.
- A Sephardi woman, according to Rav Ovadia, should cover her legs.
- The area above the knee or elbow is considered immodest in the laws of Shema, below those areas depends on the custom of the area.
- One should be careful to get dressed under one’s blanket so as not to reveal skin that is usually covered. Therefore, if one is not wearing a shirt to sleep, one should put it on under the blanket rather than sit up and put it on. 
- Similarly, in places where people walk around in socks and shoes even in the summer, if it is possible one should put on one’s socks underneath the blanket; however, some are lenient regarding revealing one’s feet. 
- One should make sure to cover one’s body at least up to the breast bone and on one’s arms at least up to the elbow. Some are lenient regarding short sleeve shirts. 
- There is what to rely on to change in a bathroom stall without being concerned about one being temporarily undressed, since it is a closed private area. 
- In a men's bathhouse where it is common to walk around naked and is impossible to bathe otherwise, it is permissible to do so.  The same is true of when one bathes in a river. When one goes in to bath one should not cover one’s private part because otherwise it looks like one is covering up one’s Brit Milah, however, upon coming out of the water one should cover one’s private part. 
- When bathing or dipping in the Mikvah, one should be careful to unclothe and get dressed as close to the water as possible. 
- Rav Hershel Schachter in a shiur on yutorah
- Yishayahu 45:15 describes Hashem as being a God who hides.
- Devarim 28:9, Shabbat 130a, Sotah 14a
- Micha 6:8 states that a person should be modest in His walking with Hashem. The Gemara Macot 24a explains that this is one of the three major principals of Torah.
- Rav Ovadia Yosef in Taharat habayit v. 2 p. 164 writes that a woman should cover her arm from the elbow and up including the elbow.
- Rav Ovadia Yosef in Taharat habayit v. 2 p. 164 writes that a woman should cover her legs even below the knees.
- Mishna Berura, 75:2.
- S”A 2:1 writes that a person should not sit up before putting on one’s shirt rather one should put it on under one’s covers.
- Mishna Brurah 2:1 writes that since in his part of the world people covered their feet even in the summer one should put on one’s socks under the blanket if possible, however, Aruch HaShulchan 2:1 is lenient.
- Mishna Brurah 2:1 writes that one should cover one’s body up to the breast bone and up to one’s elbows. However, Rav Nevinsal in BeYitzchak Yikra 2:1 quotes Rav Shlomo Zalman who was lenient regarding short sleeve shirts (see there).
- Kaf HaChaim 3:13
- Halacha Brurah 2:1
- Mishna Brurah 2:1, Aruch HaShulchan 2:1
- Otzar Halachot 2:3