Confused about blood tests

So at my first appointment my dr drew blood. This week I was 11 weeks along and they took a lot more blood. He said they'd look for chromosomes. 
First, how can the tell baby's chromosomes from my blood? Second, is this the way you find out if you have soft indicators for downs & such? Are there more tests after this?
Just wondering if anyone can explain this to me because I'm a bit sketchy on it. Thanks!!