All the governments of the middle colonies had a democratically elected legislature, a governor, a governor's court, and a court system. Although the legislature in each colony was democratically elected, the governor was not. In most colonies, he was appointed by the English king. It was therefore the king to whom the governor owed his position, not the people he governed.

In some colonies, such as Delaware and Pennsylvania, the governor would be chosen by the people, but he crucially still had to be accepted by the king. It was he who enjoyed final approval, not the people themselves.

Two of the middle colonies, New York and New Jersey, were royal colonies, meaning that they were ruled directly by the English monarch. The rest were proprietary colonies, which arose out of land granted to colonists by the king.

A good example of a proprietary colony would be Pennsylvania, which was founded after King Charles II gave William Penn a large tract of land as payment for debt owed by the Crown to Penn's father.

First, an overview:

The Middle Colonies are New Jersey, New York, Delaware, and Pennsylvania.  New York and New Jersey were ruled as royal colonies, and Delaware and Pennsylvania were ruled as proprietary colonies.  A royal colony was ruled directly by the English monarchy, and a proprietary colony was given to one or more people by the English crown but ruled by that proprietor.  William Penn was the proprietor of lands in Pennsylvania and Delaware.  

Regardless of whether the colonies were royal or proprietary, all colonies elected their own legislature, had a governor, and had a court system.   The governor of the New York and New Jersey royal colonies was elected by the King, and the governor of the Delaware and Pennsylvanian proprietary colonies chose their own governor and that governor was then reviewed by the King (the monarchy had to approve of the chosen governor).  

---

New York was founded by the Dutch and land was given to the Duke of York.  The governors were chosen by the Duke of York, and a council was created to help the governor with decision-making (the council was elected, meaning that colonists were able to give a small amount of input in the decision-making.)  The Duke of York gave some land to George Carteret and John Berkeley, and this became the New Jersey colony.  These two men lived in England and also chose their governors, who elected a council to help with decision-making (much like New York).  To reiterate, although colonists were chosen to represent in this council, they had little rights or representation in England.

William Penn was a Quaker from England who was given land in the New World.  The King owed the Penn family a lot of money, but instead of paying out his debts and perhaps to remove the Quakers from England (which was an Anglican-majority country), Penn was given a large plot of land instead.  This became the proprietary colony of Pennsylvania, where many people enjoyed religious tolerance and greater representation.  Pennsylvanian colonists could vote for their representatives who had the power to approve or reject laws, giving them more power than their New York or New Jersey counterparts.  Penn was also given a portion of land which later became the colony of Delaware.

The Middle Colonies were governed similarly to the other colonies. Each colony was governed by a governor and a legislature. Legislatures were generally elected by the residents of the colony, and they could enact laws governing the conduct of colony members. These laws could be far-reaching and even regulate public religious conduct or private behavior. The governor could overrule these laws as the governor of a colony was not elected by the residents of the colony but was installed by the King. The colonies also had court systems, though these were different than what we would expect today, as many settlements were too small to have permanent court houses and judges. Traveling could be difficult for many citizens, which led to the use of traveling courts for the settlement of some crimes and civil complaints.

The government in the Middle Colonies was, in general, the same as the government in the other colonies.  All of the colonies had the same basic governmental structure.

All of the Middle Colonies had a chief executive called a governor.  In all of the colonies, the governors were appointed, not elected.  Some of the Middle Colonies, like Pennsylvania, were proprietary colonies.  In those colonies, the proprietor got to choose the governor but the king had to approve the choice.  In the royal colonies, like New York, the king chose the governor directly. 

In all of the Middle Colonies, there were bicameral legislatures.  The upper house of the legislature was also appointed just like the governor.  The lower house of the legislature was elected by the people.  Most white men could vote in at least some elections.  Some colonies, like Pennsylvania, let essentially all white men vote for their legislatures.  In other colonies, like New York, only white men with a certain amount of property could vote in the legislative elections.  White men with less property or even no property could vote, but only in local elections.

The Middle Colonies, then, were similar to all the other colonies in that they had appointed governors and upper houses of the legislature and they had lower legislative houses that were elected by the people.