The root port is the port closest to the root bridge. A designated port is a non-blocking port. The root bridge will have all designated ports.
Here's a crude diagram to clarify:
A (root bridge)
/ \
B---C
The ports on A connecting to B and C are designated ports. The ports on B and C connecting to A are root ports. On the link between B and C, one side is blocking and the other side is fowarding (which side depends on the STP election process). The side which is forwarding is the designated port.
