NNI stands for Network to Network Interface. This is a frame relay connection that is used to link different carrier's networks to each other. For example, Carrier A might need to point some PVC's to routers on Carrier B's network. Carrier A would need to be connected to Carrier B in some form (T-1, DS-3) in order to pass traffic between each other. An NNI is is a pipe between networks.
UNI stands for User to Network Interface. This is the port on a frame relay network that a customer hooks a location into. A UNI port is the customer port(s) in a frame cloud.