Preamble to the United States Constitution
"Establish justice" meant to establish the rule of law. It meant that everyone was equal before the law with no distinction between the status or wealth of a person. With their new government, the Founding Fathers wanted to establish (create) justice and fairness.  When the British Empire governed the American colonies, the colonists did not think the government was fair.  For example, Parliament would decide that the colonists had to pay taxes on items like stamps and things that were made in other countries.  The colonists did not have any say in this.