A lot of the weird things about BYU actually directly relate to weird things about LDS (Mormon) culture.
BYU is 97%. Everyone assumes that everyone on campus is Mormon, which is true most of the time. This can cause problems though. For example, my friend crossed herself after a prayer and some girl yelled at her. It is also often assumed that all people on BYU campus are Republicans. The exact ratio is unknown, though presumably non-Republicans are greatly in the minority. However, it's large enough that back in 2007 when Dick Cheney came to speak at BYU commencement and several large protests ensued, eventually bringing Ralph Nader to an "alternative commencement."
They pray at the beginning of classes. Not just religion classes, but depending on the teacher, also math, science, music, etc. It's not required by any means but it's pretty common. It is also not uncommon to hear teachers bear their testimonies, especially at the end of the semester. Some think this is because there is a question about spirituality on student ratings of teachers/classes.
They also pray before sporting events as well as a lot of other events and even club meetings. Basically, if you're going to an event at BYU, there will most likely be a prayer, probably just at the beginning but sometimes at the end as well.
Mormons are freaking awesome
BYU is made up of mostly conservative Caucasians. A third of students are from Utah.
There really aren't a lot of men ages 18-21 on BYU campus, because most of them go on two-year missions. Men are strongly encouraged to go on these (they won't be excommunicated if they don't or anything), and women can go if they want to. Women cannot serve missions until they turn 19 and theirs only lasts a year and a half. Men cannot go on missions after they turn 26 but women can go later. Old married couples also go on these. Because so many BYU students go on missions there are a lot of people that speak other languages, sometimes pretty random ones such as Tagalog, Lithuanian, Quiche, etc. Also, BYU students, particularly people that have served missions, are heavily recruited for things such as selling security systems door to door, as well as by prestigious government agencies, who want to capitalize on their proven work ethic and language skills.
The BYU Honor Code = more strict than LDS standards.
no facial hair (excepting mustaches of a conservative non-Tom Selleck nature) or long hair on guys long shorts are the norm and leggings with shorter shorts/skirts are apparently risque curfew - on fridays people are allowed to hang out with the opposite sex in their apartments until 1:30 am. otherwise, it's midnight. no people of the opposite sex in your bedrooms ever. that goes for the dorms and also off-campus housing. also, if possible, they shouldn't use your bathroom unless it's an emergency. men (presumably minus the swim team) are not allowed to wear Speedos in campus pools or BYU contracted housing. women are not allowed to wear bikinis.
No fund-raising on campus, except during "Care Week" which happens once a semester (not sure if they exist during Spring/Summer terms, I don't think so). Sometimes, during winter semester they will have Care Week be the same week as BYUSA elections, which means that everybody is being bombarded by ridiculous campaign antics so they don't want your flyer about helping poor people. However, if a club wants to raise funds for a trip or something of that nature, they can do that year-round.
No spring break
If you have a beard or are wearing a short skirt you tend to get stared down. People will assume you go to UVU or are visiting or something.
BYU is owned by the LDS church. It is largely subsidized through tithing paid by its members (1/10 of their income). Because of this, people that are members of the LDS church pay less in tuition than people that are not.
Probably at least 15 times over the years I have walked past the Eyring Science Center toward the Spencer W. Kimball Towers and smelled marijuana. Maybe there's a secret smoking area around there? Or maybe they conduct research on the effect of marijuana on rats? I don't know but it's weird and freakin nuts.