1. If you are speaking about yourself and your family:
* You and your family are invited. (This is the correct way to say it if you are including yourself in the invitation)
2. If you are speaking about someone else and their family:
* They and their family are invited. (This is the correct way to say it if you are referring to a third party)
Explanation:
* The pronoun "you" is always used in the nominative case (subject of the verb) when referring to yourself.
* The pronoun "they" is also always used in the nominative case when referring to a third person plural.
Therefore, the correct way to phrase the sentence depends on whether you are including yourself in the invitation or referring to someone else.