I would say that one way the Senate has been important is through its power to conduct investigations into abuse of Presidential power. We need only look at the Watergate Hearings as an example of this. At this historical instant, the Senate was the body that was able to shine light, bringing much needed illumination, to the dealings in the White House and the Nixon Administration. Senator Ervin's hearings allowed a nation to fully grasp what their President was doing and what passed for Executive Leadership at the time. When Senator Howard Baker asks the questions, "What did the President know and when did he know it," is was a moment in time when the Senate was able to capture the essence of questioning that many Americans felt, proving that for an instant, the Senate can truly represent the will of the people.
If you mean the Senate in general over the course of US history, then it is a very important part of our government. Treaties can only be ratified by the Senate, and it also is in charge of approving all nominations for federal judgeships, Supreme Court appointees and military leadership posts. In this way, it is an important check and balance against the power of the President. He has to work with them to get what he wants done.
In addition, the Senate is one half of the lawmaking body we call Congress (with the House of Representatives being the other half). In order to become a law, a bill has to pass both the House and the Senate with a majority vote. If it only passes one of them then it fails and is shelved.
The Senate also held the impeachment trials for Andrew Johnson, Richard Nixon and Bill Clinton.
Throughout the history of the US, the Senate has been important for a few different reasons. I will touch on three of them.
During the "Gilded Age," the Senate was important because it served as a brake on democracy. The Senate was not elected by the people and so it was dominated by the interests of the rich. This helped prevent reform from happening.
After WWI, the Senate was important because it checked Woodrow Wilson's attempts to get the US into the League of Nations. This seriously affected US foreign policy.
Today (and for the last few decades) the Senate has been a major player in slowing down legislation. The minority party has used the filibuster to prevent legislation it does not like. At other times, the majority party (if different from the President's party) has used the Senate to oppose the President's policies.