There is a lot of buzz about Senator Bernie Sanders becoming the Chair of the Senate Budget Committee. The question is, though, what powers does he actually gain when he takes up the Chairmanship?
Effectively, the committee chair is the top official in the Senate with respect to that policy area. And, in the area of the budget, how the budget process is framed and managed can heavily impact the outcome.
The chair has the power to set the agenda of the committee and to preside in its meetings.
The first power is the most important.
When a Senator introduces a bill, or a bill is passed by the House of Representatives and sent to the Senate, the first thing that happens to the bill, which is called a "first reading" is for it to be referred by the appropriate person (usually the presiding officer or the rules committee) to one or more committees. Often a bill is referred both to a subject matter committee (e.g. the transportation committee) and a fiscal committee that approves funding for it. (This said, a budget is just a roadmap for spending that frames the appropriations process, the actual decision to spend is made in separate appropriations bills.)
The chair of a committee decides which of the many bills referred to it to consider, when to consider the bill, and for how long. A chair also handles the determination of which bills are referred to which subcommittees of the committee.
Absent special circumstances, a bill will only be considered by the full Senate if the chair schedules the bill for a vote on to recommend adoption of the bill by the Senate and a majority of the committee approves it.
A chair who has the support of enough committee members can kill a bill indefinitely by refusing to consider it. The chair can also decide who is recognized to speak on a bill, prioritize which proposed amendments to a bill are considered in what order, and has the primary role in deciding who will be called to testify before the committee.
The chair also has a quasi-administrative role as the primary supervisor of committee staff who hires them and oversees their work, and as the person in charge of controlling the committee's office space.
Since budget bills affect almost everything in government, and budget bills are considered "must pass" bills, the budget bills referred out of the budget committee will almost always be the starting point of floor debate on a national budget of the United States government in the Senate that will ultimately (possibly with some floor amendments) be passed by a majority of the U.S. Senate (something not true of bills from many other committees with narrower jurisdiction).