This answer is going to be long...
The first part, when you're about 10-13 weeks, is a "first trimester screen". This can either be a standalone screen for first trimester, or you can combine it with a screen in your second trimester for more accurate results.
The goal of screening (note: SCREENING. This is NOT diagnostic and will only let you know about increased risks, not yes/no.) is to detect pregnancies at higher risk for abnormalities, specifically Down Syndrome and Trisomy 18.
The first trimester screen usually starts with an ultrasound. The sonographer will take measurements of the baby, including nuchal translucency. That's the amount of fluid at the back of the baby's neck (and it's normal for the baby to have fluid). If it's above a cut off, the geneticist or a genetic counselor will come and talk to you about what it could mean and your options (i.e. offer diagnostic testing, such as a CVS [chorionic villi sampling] or an amniocentesis [amnio]). You would not have your blood drawn for the blood analysis because the screening has done it's job and identified increased risk.
If it's in the normal range, you'll have your blood drawn. The lab will look at amounts of chemicals in the blood that are produced by your pregnancy, your age, and the nuchal translucency score and give you a risk number.
The risk number will be a 1/something. The lab we used ranged from 1/5 to 1/10000. Usually when patients are given their results, they're told what their risk related to age alone was first.
The screen in the first trimester is 80% accurate. That means 8 out of every 10 babies that do have Down Syndrome (DS) or Trisomy 18 are put into a high risk category. The lab we used, that was 1/49 and higher (1/48, 1/47...). So, a >2% chance of having a child with DS was the cutoff point (for reference, the chance of having a child with abnormalities of some kind is 3-5%).
If you have the sequential screen, which has a blood draw in the second trimester (16-18 weeks), the lab will analyze that blood, and combine the numbers with scores from the first trimester screen, and give you a risk number. It's considered 90% accurate at this point because of combining the two screenings.
Of the women who receive "high risk" numbers, 95% of them do NOT have a child with a chromosomal problem. But they've been caught in the balancing act the lab has to play to set the levels where they'll detect the most pregnancies at risk for those disorders without detecting too many that aren't.
Many doctors will tell women that they "screened positive" for DS without explaining what that means. Screen Positive = Increased Risk (above a cutoff). Not diagnostic! And so many women think their baby has DS, and then when they have the amnio or CVS and it comes back normal, they're like, "the doctor was wrong!". It's possible it wasn't explained to them clearly before (or after) they had the screening, or at the time of giving results. Or that they forgot everything the doc said after the words "screen positive for down syndrome". Who knows?
Anyway, it is important to realize that it's only a risk. If you are not comfortable with the number you hear, you can have diagnostic testing. It's also paramount to think about what you'd do with the information before going in for the diagnostic testing, which most of the time can tell you yes/no (because it's looking at the baby's chromosomes directly).
Are you going to use it to make a decision about carrying/terminating the pregnancy? Or use it to prepare? Sometimes children with chromosomal abnormalities have heart issues or other issues, and this way, the parents can ensure that the right professionals are on hand at the time of birth to help their baby.
CVS is done around 11-13 weeks and samples the cells from the placenta. This works because the baby and the placenta had to grow from the same original egg and sperm, so theoretically, have the same chromosomes. Very rarely they don't (<1%, IIRC).
Amnio is done anytime after the 16th week (where I was, anyway). It's a long, thin needle (it's smaller than a blood donation needle) inserted into the amniotic fluid surrounding your baby to get a sample that will go to the lab. The needle would not be in for more than a minute (if you've a skilled OB/GYN to do it) and most women say it hurts less than they expected.
Biggest risk is miscarriage. The old risks quoted were about 1/200, but now they're better than that. At the hospital I'm at, it's about 1/300-1/500 for a miscarriage for a CVS (and that risk is higher because miscarriage is more common in the first trimester anyway), and 1/500-1/1000 for amniocentesis.
So...armed with all that... the choices are up to you. I think some people don't want to do it because they would not terminate the pregnancy, and that's totally fine. Some don't want to do it because they feel the fear of miscarriage is too great. Some do it because, for their own re