Temple Garments
One of the unique questions revolving around the Church of Jesus Christ of Latter-day Saints is in regards to “Mormon underwear,” known properly as garments. It has even made the news a few times - you may recall how this came up in Mitt Romney’s presidential run as he was asked by reporters to comment on it.
While it’s admittedly an odd topic for others to ask about, it is indeed a question that comes up a lot. Hopefully I can shed some light on the subject and encourage a better understanding of the background and context for the temple garment.
Throughout history and into the 21st century, religious worship has often carried with it a distinct expression of devotion through clothing. Below are a few historical and modern examples.
Aaron and Ceremonial Clothing
In Exodus 40 Aaron is dressed in sacred, symbolic clothing by Moses. This clothing carries with is specific meaning and reminders as Aaron officiates in priesthood ordinances.

Priest’s Garb or Vestments in the Episcopal Church
"Many of the church's vestments are descended from the ordinary dress of the imperial Roman society in which the early church came into being." (See link above)

A Jewish Shawl or Tallit is worn as a reminder of the duties and obligations of a Jew. It is worn to fulfill biblical precepts regarding clothing requirements.

The variation of religious vesture throughout the world is a beautiful thing to contemplate and to behold. There is a rich tapestry pointing toward our devotion to a higher power, and a dedication to religious principles. This woven fabric can be seen across the Virginia Peninsula as we are fortunate enough to be house many of these religions in our community.
The articles of clothing are worn as an outward expression of an inner commitment to adhere to promised principles. The temple garment fulfills the same purpose for members of the church.
To further understand the purpose of the temple garment within the church, it is important first to understand temples. Temples are more than just chapels. They are holy edifices in which we make promises that are sacred to us - promises that help us lead a more Christ-like life and ultimately return to live with our Father in Heaven. The garment stands as a daily reminder of those sacred promises we’ve made and why it’s so important to keep those promises.
“Mormon underwear” or “magic underwear” is, in general, an attempt at a comical moniker. However, members of the church regard the clothing as sacred, and it is much more respectful to refer to the them as garments, or temple garments.
Overall you'll find that inquiring about this topic with members is not offensive at all. We understand the curiosity surrounding it and are happy to talk about why we find it so important to adhere to this principle of our religion. Once you understand the importance of Temples in the church, the temple garment is a natural offshoot of those beliefs.