In a lot of ways the Senate did the same things they did in the Republic. It was a group of the most prestigious men in Rome who gathered to advise the people on issues of public importance but did not actually legislate. In the empire, the People was embodied in the emperor (Augustus and his successors liked to highlight there tribunal powers on their coins more than their other titles which shows that they thought the link between them and the people was integral to their office).

The Senate also controlled some provinces (Africa, the northwest coast of the continent, being the most important). They decided who governed those provinces and in Africa the governor even had an army to command.

Despite these powers it should be noted that the Senate was under the emperor's thumb. Tacitus talks about how the senators based their policy on trying to gain the favor of the emperor, and the emperor also had the power of the censor, the magistrate who had the responsibility of adding men to the Senate and removing them if they lost their social standing or moral authority.

Augustus went to great lengths to look like he was consulting the Senate for all decisions he made. This is because Rome had a pathological hatred of tyrants, from early in their history. So Augustus went through the motions, very diligently, through is long life and reign. This way, form was preserved and the people would not get upset enough to overthrow Augustus.

Is it true that the Senate ended up with the privilege of issuing the "judgment" of history, that many stories and manuscripts about imperial reigns came from their ranks? The nicer an emperor was to the senate, the more positively they remembered him. So someone like Hadrian got a good-guy edit, even though they could have made him out to be a poltroon with selective editing.

Can't remember where I heard/read the observation, sorry...