Tznius is most often translated as "modesty," and people usually use it in conversation to refer to women's clothing. However, most authorities I've come across seem to agree that it's really an attitude. It is also one of the hardest words for a new-to-Jew to pronounce.
I highly suggest reading Outside Inside by Gila Manolson. It's a short (and cheap!) overview of the topic. Though it can be a little too "teen" feeling for some people, I think it's a great introduction to the ideas of tznius. You can easily read this on a Shabbat.
a) Men and women both must wear a minimum amount of clothing when in public. What those halachic minimums are may differ based on who you talk to. Community standards certainly will differ, sometimes even down to which street within a community you're talking about, lol... That said, most of the orthodox world will agree that
- Men should wear knee-length shorts, a t-shirt (therefore a high neckline), and a headcovering.
- Women should wear a skirt slightly past knee-length (most agree on this as a minimum, but it's debated heavily), a shirt that reaches the elbows (most say covers the elbows) and has a high neckline (some say collarbone, all say no cleavage). Once married, some headcovering will be required at least in synagogue (most say full-time, and all argue about how much covered and how covered).
c) Being shomer negiah (not physically touching the other gender) can be classified as tznius, though I personally classify it as part of the laws of family purity. It just makes more sense to me that way.
d) Being careful what information you put into your head. This means movies, music, books, magazines, whatever. This is an area where people start to get ruffled in "tznius" conversations because it almost inevitably leads to one person feeling "judged" by the other. Some people refuse to have televisions or the internet in their homes because they're "not kosher," but they really mean they're trying to avoid the "non-tznius" aspects of those media (remember, media is the plural of medium!).
e) It's an easy way to justify hating something you don't like ;)
Tznius is a HUGE issue in the orthodox world. Really. Especially when you limit the definition to women's clothing. For more proof, see my recent post Orthodox Women Being Patronized by Feminists? and its comments. Personally, "It's just not tznius" is usually a sign to me that a conversation/lecture/article is about to get really annoying really fast. The idea and halacha have greatly improved my life, but they're also greatly abused to justify all kinds of statements.