The political structure in the New England colonies was different at different times in colonial history. There were also multiple levels of government.
At the lowest level, New England government was centered around town governments. These governments, early on, tended to be very closely tied to the Puritan church leadership. The local governments tried to maintain their independence from the larger colonial government.
On the colonial level, governments tended to be partly elected and partly appointed. They had a governor appointed by the British government. They had a legislature with one house that was appointed by the governor and one that was elected by the people.
The New England colonies were founded by people who left England to escape religious persecution. These were mostly Puritans who also had strict beliefs of their own about proper conduct, and they disapproved of many customs and behaviors that their English countrymen enjoyed. The English were happy for the Puritans to leave, so as to stop hearing them complain, and the Pilgrims, as they called themselves, embraced the opportunity to start anew in their own place, despite the hardships they would face.
Without a monarchy and an aristocracy to impose laws, and without a lower class to assign tedious tasks, the middle class found itself in a perfect situation to practice democracy. The men were responsible for providing for the families they brought with them, and they created political structures that served their needs. Within the colonies they formed townships that governed the activities of the inhabitants.
The citizens of each town took seriously their personal responsibility to participate in the governance of the town. They did not elect representatives. They participated in the discussions, and voted for the projects and the taxes they would pay-- themselves. They provided for record-keeping, and appointed clerks. They wrote laws to prescribe acceptable behavior and outlaw what they deemed to be evil. The townspeople elected magistrates to preside over courts to settle legal disputes and impose punishments, many of which would be considered cruel today. But it was what the people of each township wanted for themselves. Laws were not imposed by any sovereign monarch, but based on common law and what they believed was their duty before God to have His will-- as they understood it-- be done in the land God had given them.