Mitochondria do indeed have their own genome, as well as their own machinery for making RNA and proteins from the genome. There are additional proteins that are synthesized from the nuclear genome and are imported into the mitochondria. As a result, it's the synchronization of mitochondrial and cellular replication that is the key point to understand (and it is not yet understood).
Alberts et. al "Molecular biology of the cell" has extensive material on the subject in chapter 14. It's a fascinating topic that relates to the origins of aerobic life.
However, mitochondria do not have a membrane-bound nuclei. Their transcription and translation resembles these processes in bacterial cells, which also have no nuclei.