For the most part, the powers of Congress are formal powers. The Congress is the only branch that has the power to propose bills and to pass them so they can become law. The Senate (one branch of Congress) has the power to confirm or reject people that the president nominates for the Supreme Court or for various other high government offices. Both houses together have the power to override presidential vetoes and to propose constitutional amendments. These are all very formal powers and it is the formal powers that make Congress extremely important in our system.
Congress (or individual members of Congress) has very little in the way of informal powers. Congress has some informal power over the president’s agenda. A president might change his (or someday her) agenda because he knows that Congress will never pass the laws he really wants. The same is true of people that the president appoints. President Obama nominated a fairly moderate person to the Supreme Court after Justice Scalia died, knowing that the Senate would never confirm a true liberal. Thus, the Senate affected Obama’s choices in an informal manner.
Individual members of Congress can also have some informal powers if they are sufficiently well-known. A good example of this right now is Paul Ryan. Ryan is the Speaker of the House and was Mitt Romney’s running mate in 2012. He is well-known and fairly widely respected. He is trying to use his high profile to push the Republicans’ policies in the direction that he wants. He wants to do more to focus on conservative policies that will help the middle class and the poor. If he can get enough people to pay attention to his speeches, he might be able to exert some control over the government’s agenda.
These are the main types of informal powers held by Congress and by members of that body. Congress can use its formal powers to give it informal influence over what presidents do. Individual members of Congress can use their high profiles (if they have high profiles) to try to steer the national agenda in their preferred direction.