In the United States, the executive branch of a state government is the office of the Governor. As a citizen or resident, you can influence the governor of your state in several ways.
The single most powerful tool you have at your disposal is your right to vote, which enables you to participate the the process of choosing the governor for your state. Your vote counts in two stages of the election process, the primaries in which the two major parties select candidates and the actual election. If you want to get more deeply involved in the political process, you can join a political party and become active in campaigning.
Two other ways to influence the governor of your state are enshrined in the First Amendment. On the state as well as the federal level, you are guaranteed the right to petition and the right of peaceable assembly. This means that you can gather signatures and submit petitions to the governor (or any other politician) and organize protest demonstrations. You can also send paper letters or emails to the governor.