Your family is your direct relatives, whereas your dynasty includes all your relatives (Example : You live in France, you have a specific last name, and there's people you know nothing about in Germany who have the same last name as you? They aren't part of your family, but they are part of your dynasty).

As for a Household, the difference between that and a family is that the Household includes those who live with the family but aren't part of it, too. Think, for instance, of a large noble house where there are servants and constant bodyguards; those servants and bodyguards are part of the household, but not the family (A great example is in Skyrim, in Whiterun, when you're given Breezehome and suddenly "Lydia of Breezehome" starts tagging along as a companion; Breezehome is a Household, you're part of it but Lydia, with whom you have no blood connection, is also part of it).