The cervix is the organ between the uterus and the vagina, and it serves as gate for expelling menstrual fluid (and for letting the baby out during childbirth). This word comes from the Latin word that means neck, and in many languages it's called the neck or mouth of the uterus. You can find out more in Wikipedia, and if you prefer to get this information in a language other than English, you can most likely find the equivalent term there as well.
There are many guides to finding your cervix, both here and on fertility-related sites, so this will be covered very briefly. The most common description is that it feels like the tip of your nose; a very low/dangly cervix can also be similar to a small penis. There's a small hole in the cervix, and this is where the blood comes out.
Note: the hymen is not in the middle of the vagina. It does have a hole, but it is normally large enough for inserting a finger, folded cup or tampon; the cervical opening is MUCH smaller. So if inside your vagina you can feel something with a hole, that's the cervix.
See also: Beautiful Cervix Project (graphic photos), Melissa's video, very NSFW photos of the hymen and the vaginal opening; glossary.
How low is low?
Determining the cervix height is not an exact science. If you suspect something might be abnormal, find out more about prolapse, but note that the range of normal is huge. This also means that if you ask a doctor about your cervix position, you need to specify that you are trying to compare your anatomy to other people's.
In the context of cup use, mild prolapse is not different from having a naturally low cervix. The former can be caused by pregnancy/childbirth or age (but the manufacturers overrate the importance of these factors), the latter is just anatomical variation that can be permanent or only occur on some days of the cycle, especially the heaviest days of your period. Your height or weight have nothing to do with the menstrual cup dimensions you require.
Let me also point out that the name doesn't matter and there's no standard scale for companies. Just because a cup is called "medium size", "low cervix version" or marketed to virgins, doesn't mean it's actually better than the other options. Choose by the actual dimensions! The good news is that cups aren't like bras or shoes. It's not about finding the one correct size. (this part has been added later)
I propose the following definitions for the cervix position. They are left deliberately vague, with no numbers given. In the classic cases, all parts of the description apply, but of course this can vary if your fingers are unusually short or long.
high: The cervix is unreachable or can only be reached in certain positions. Likely to need bearing down for removal. Long cups fit best, although a long stem can be enough.
low: The cervix is easily reachable, less than a cup's length away. Has to let the cup unfold as soon as it's halfway in and/or use only the smallest cups. Likely to need to trim or cut off the stem.
extremely low: The cervix is very easy to reach, it's immediately in the way during insertion. Normally, it's needed to let cup unfold as soon as the rim is in (or even part of the rim) or, alternatively, maneuver the cervix into the cup after it opens. The stem generally sticks out, so does the bottom of most cups.
average: None of the previous descriptions applies fully. Can use most cups, apart from the really long and really short ones. Most special needs or preferences have little or nothing to do with the length.
Notes: Insteads are completely different from the bell-shaped cups, but if you can use them successfully, your cervix is probably high or average. The opposite is not necessarily true - if you have difficulties with Insteads, this doesn't automatically mean your cervix is low.
For the purpose of this scale, contraceptive diaphragms are similar to Insteads and cervical caps are similar to the smallest cups on the market. However, please don't rely too much on your experience with contraceptive devices. And definitely don't take it as a suggestion to use them for collecting menstrual flow.
More about the dangly cervix
The cervix is not (necessarily) the end of the vagina, and finding your cervix is not the same as figuring out how deep you can reach. The distinction is not always important or obvious, but when the cervix dangles into the vagina, it can seem lower than it really is, and if you use a menstrual cup it can reduce its capacity, physically causing it to overflow sooner than expected.
The tell-tale sign of a dangly cervix is having a cup that works normally for a few hours, but then "leaks" (actually overflows) when it's about half-full. Depending on the individual circumstances the actual capacity available can be as low as 25% or as high as 80%. If you have a heavy flow, this can be quite hard to notice. Cups with a rounded bottom are generally recommended for maximizing the capacity you can have. See the highest capacity cups here.
Two cases merit a special mention. An extremely low cervix is typically also dangly. It's a combination of this factor and a short vagina. When the length of the vagina is about average, it can be hard and unnecessary to figure out whether your cervix is average and dangly or just low. However, a high dangly cervix (and/or dangly cervix, long vagina) can be the cause of unexplained leaks, since it's harder to figure out whether your cervix is dangly if you can't feel it.
Since this is supposed to represent the common knowledge, let's have a poll! I hope it's clear enough that this only concerns finding the right length and not other factors like capacity, stiffness etc.
Do you agree with the proposed definitions?
Cervix demography! According to the definitions, your cervix is:
If you misclicked or misread something, you can change your answer here.