There were two major ways in which the structure of government changed during this time period.
First of all, there was the change that occurred as the colonies became states in an independent nation. The major change here was in the fact that more state officials (as compared to colonial officials) were elected by the people. In colonial times, both the governor and the upper house of the legislature were typically appointed by the Crown. After independence, all parts of state governments were usually elected.
Second, there was the change in the structure of the national government that came about with the writing and ratification of the Constitution. Under the Articles of Confederation, there was really no executive branch to speak of and the national government had very little power relative to the states. Under the Constitution, there was a separate and relatively power executive branch as well as a judicial branch. (I should mention as well that the Congress under the Articles was unicameral where the Congress that the Constitution set up is bicameral.) The new national government was also given more power. For example, it had the power to impose taxes on the states and to regulate interstate commerce.
These are the most important changes in governmental structure during the time that you are asking about.