If you plan on using the placenta within the first 48 hours than you can keep it in the fridge, otherwise, put it in the freezer. And Yes... you can keep your placenta... it is yours!!! You may just have to sign a paper/release form.
I told my doctor ahead of time that I wanted to keep my placenta and encapsulate it. I reminded them at the hospital also and had them tell what they did with the placentas. At the hospital I was at they refrigerate them for so many days then dispose of them if they do not need to analyze them. After my baby was born (yes I constantly bugged the nurses about the placenta) they brought me the release paper signed by one of my doctors as soon as I was put in my room from recovery. My husband took the paper to pathology and picked it up that day. They had it in a biohazard bucket like tub with a lid for us with my name on it.