Halacha in a Nutshell is a series that does not aim to actually teach you halacha. The goal is to acquaint you with the general ideas of a halachic issue so that you can follow conversations without looking like a total n00b.
Now here's a fun topic full of vague euphemisms. This will not be comprehensive in any sense of the word (it's a pretty technical area), but hopefully you'll get the general idea. I bring this topic up because the laws of family purity are referenced to and joked about quite commonly, particularly among women, but almost always in a very vague way. If you don't know the basics of the family purity laws, you will have no idea what someone has just said to you because it was such a vague euphemism.
As a sidenote, the laws of family purity are generally not practiced by liberal Jews. You will very often hear jokes about the laws of family purity in liberal Jewish circles. In orthodox circles, it is usually not considered an "appropriate" area of public discussion, and no one should ever ask details about your personal observance of these mitzvot. If they do, feel no shame in telling them that you don't think it's an appropriate discussion to have or change the subject. This is a very personal mitzvah and is no one's business but your own (and maybe your rabbi's).
Known in Hebrew as taharat hamispacha, the laws of family purity govern the sexual relationship. It is presumed that this is a relationship between a married husband and wife. Because people aren't supposed to have "relations" before marriage (and you wondered why frum kids marry young!), this area of the law generally isn't taught until someone is engaged. In fact, several people were shocked to see it required in some detail on a conversion syllabus for a well-known American beit din. The idea is that women are taught this area by the kallah teacher (kallah means bride) when a girl is engaged. I'm really not sure when men are taught this, and my suspicion is that they're given the rudimentary details by a rabbi during engagement and are then told to trust their wives. Whether it's right or wrong, this (like kashrut) is an area where women are given significant leeway to make their own rulings in questionable situations.
At its most basic, there should be no "relations" (sorry, I feel silly saying that) during the woman's menstrual cycle and the 7 days after it ends. The days of the menstrual period are called "red days," and the days after are called the "white days." The red days last the longer of your cycle or five days. So if your cycle is 9 days, you have 9 red days. However, if your cycle lasts 2 days, you still have to count 5 days before you can count the 7 white days. This is the period called "niddah." If you say a woman "is niddah," it's during these red and white days. Niddah is usually translated as "unclean," but a more accurate translation is "ritually impure." That isn't a value judgment that women are impure, anything can potentially be ritually impure (and all of us today are ritually impure for the purposes of entering the Beis HaMikdash). This mitzvah is the source of a lot of battles with feminism, but both sides are arguing past each other because they're not talking about the same issue.
For couples who are shomer negiah, this means that they are shomer negiah to each other (and with some additional restrictions according to their minhag) during this time. So to spell that out, in addition to no "relations," they also don't touch each other, hug/kiss each other, etc. This can also extend to passing each other items, eating from the same plate, or even looking each other in the eye. An example often joked about (though a totally serious halachic ruling) is not playing tennis because that is passing the ball to each other. As a practical matter, this means that the spouses are more restricted with each other than with other members of the opposite sex. As you can imagine, this can be a very difficult mitzvah to keep! On the other hand, couples say this means they have to be especially careful to develop their ability to communicate. You can't end an argument by distracting yourselves with make-up sex. More common is to separate the beds during the niddah period. Most people have 2 twin beds that they can push together as one bed after the niddah period. However, twin beds aren't necessary. You can keep your full, queen, or king size bed so long as you can fit two of them in your bedroom!
In order to count the white days, the woman performs an internal inspection with a tampon or a "bedikah" cloth (yes, using the finger). This is where things get technical. She has to examine the cloth or tampon by daylight, and any sort of discharge (internal on the cloth or external on the underwear) has to be examined carefully to determine its origin because some discharges count for red days and some don't. For instance, blood from an injury doesn't start niddah (for example, from a gynecological exam - though an exam could also cause niddah bleeding). As you can imagine, this can get very complicated, and a trusted halachic advisor is essential to the proper observance of this mitzvah. Because this can be a very embarrassing discussion, more and more women are being trained as halachic advisors in this area. After all, you may have to show the person a stained tampon, underwear, or bedikah cloth. I'm sure you get used to it (and it doesn't come up all that often for the average woman), but the first few times will be unavoidably awkward for 90% of us.
Once the 7 clean days have passed, the woman can go to the mikvah, which will end the niddah period. If you want more information about preparing for the mikvah, read The Conversion Mikvah Visit in a Nutshell because the same supplies and preparation applies. There are just less dips in the mikvah and one less blessing. It is against halacha to unnecessarily delay going to the mikvah, which is something I find very interesting. After all, a wife could use that as a weapon against her husband, which isn't fighting fair.
Financially, there is normally a "recommended donation" for using the mikvah, though there may be a set fee. Every mikvah can and will be different. Some will be prettier than others, some will be cleaner than others, and some will be more crowded than others. You may or may not need to make an appointment.
Some say that the woman can't/shouldn't bathe off the mikvah water until she's had "relations" with her husband.
Here are some awesome first-person discussions of the mitzvah of mikvah:
My Mikvah, My Mitzvah
The Mikvah Is Lost on Me
The Ladies' Club (About a bride's first experience at the mikvah)