The Imperial Senate held relatively little power, and was mostly a facade. The senate solely existed to give the illusion of democracy to the citizens of the empire. In reality, the Emperor was completely in charge, though in both Canon and Legends, the senate held a little bit of power.
In canon: Essentially nothing. While we know that they could pass laws, it was the Emperor who had the final call, and rarely saw the senate's ideas through.
In Legends: Nothing Major. The senate had influence of the imperial budget, the judicial system, the legislative system, and possibly the military. The executive power held by the republic senate was stripped away. There were some laws passed by the senate, but the Emperor still had total control.
However: senators such as Leia Organa and Mon Mothma had some diplomatic immunity, which when combined with their connections to other political leaders, meant that they were in the perfect position to help the rebel alliance. In some sense, the oppressed senate helped form and guide the rebel alliance.