What is an apheresis blood donation; what are they taking from the donor?

One or more products. Apheresis means "to take away" in greek. Automated instruments can collect specific components from donor's blood. The most commonly collected is platelets, but also plasma or even red cell can be collected simultaneously. The advantage is that donors can choose how they want to contribute in saving lives. If you decide to give platelets or plasma you may give up to 24 times in a year.
Platelets. Platelets are tiny particles in the blood important for preventing bleeding. There is not much platelets found in a single bag of blood donated by phlebotomy. Most platelet donations are done by apheresis where the donor is connected to a machine where his blood is run through and only platelets are removed and the rest of the blood components (red, white cells, plasma) are returned.