"Cord blood" is the blood found in not only the umbilical cord after a birth, but the placenta as well. This blood contains many stem cells, which in turn can be used to treat a number of hematopoietic and genetic disorders.
This blood is taken for storage immediately following a birth (either vaginal delivery or cesarean section) using one of a few different options, but the practice is considered quite safe when done properly regardless of the specific method used.
The collection of cord blood always takes place after the birth of the baby, but following that can be done both in utero or after the placenta has been delivered (in utero being more common).
The most common way of doing both procedures are similar; the umbilical cord is clamped and severed at the appropriate time (which is done as part of a normal birthing procedure) and the blood is either:

or
Shown Right: Photo of a healthy placenta with umbilical cord attached taken soon after birth.
Because the collection of cord blood is not a standard practice at most hospitals or home births, a kit is required in order to properly collect and temporarily store the blood until it can be further processed. This kit is normally provided by the storage facility that the collector has chosen, or by the recipient of the donation should it be given for research. Either way, it's important for those who wish to bank cord blood to make arrangements for the necessary equipment beforehand.
The average birth results in the ability to harvest approximately 75ml of cord blood, though that amount can vary from birth to birth and with other factors such as the efficiency of the collection.
With few exceptions, cord blood that is collected is usually designated to be transported and stored soon afterwards. This is done through either a private blood bank or as a donation for research or to help others.