There are three things that are usually done in the hospital. A lot of these are done out of the hospital too, because they are the policy of the state. Note: Even if something has to be done, you always have the right to decline it. I know some but not all the stats - I think Henci Goer covers these in her book. My midwife is very anti-intervention, so she's given me a lot of alternatives to what is typically done.
1. Eye goop aka antibiotics - This is a thick liquid that is put in the baby's eye very soon after birth to prevent infection from bacteria in the vagina. Unless you have an STD, the baby's eye getting infected shouldn't be an issue. Also, breastmilk works just as well as the eye goop. The reasons mothers decline this is because it blurs the baby's vision. Some moms choose to delay this for a few hours until after bonding and nursing time. Most likely I'm going to decline this completely, and use breastmilk if there is a reason my midwife is concerned about an infection.
2. Vitamin K shot - given to the baby soon after birth to promote blood clotting in case there is a clotting disorder. I don't know a lot about this, but again I'm going to decline this unless there is a lot of bruising on the baby or other signs he/she may have a clotting disorder.
3. Heel stick - I believe this is done soon after birth then again somewhere between 2-14 days after the birth. The blood drawn is used to test for rare but usually severe/deadly disorders in the baby, most of which do not show signs for several months. I think I'm going to combine both heel sticks into one, and do this somewhere between 7-14 days after the birth. There are also ways for this not to hurt the baby as much, such as warming up the foot and doing the stick while nursing which I'll probably do as well.