A dynasty is when one family rules a country or region over a long period of time. Generally, the head of the family will be the ruler of the land, like an emperor or king. When that ruler dies, another member of the family will take power, usually the oldest son. When a new family takes control, then a new dynasty begins.
The Mandate of Heaven is what the Chinese people believed gave their rulers the right to be king or emperor. It meant that the gods had blessed that person with the right to rule. A ruler had to be a good and just ruler to keep the Mandate of Heaven. When a ruler or dynasty lost power, this meant that they must also have lost the Mandate of Heaven.