He is a CAVALIER: we can understand it by his way of dressing: he has got weapons, a big hat with feather, boots with spurs. We can imagine that he is fighting in a tournament.
It represents the Charles I Stuart execution, in 1649: Charles's trial was a public trial. On 26 January, the Commissioners drafted the sentence, condemning Charles Stuart as a "tyrant, traitor, murderer and public enemy to the Commonwealth of England". In Charles's execution Charles's face was not covered.
It is a classical example of a Puritan's family: the house is as modest as possible. They are wearing very simple dresses. The woman is covering her head as a sign of humilty.