Originally, senators did this by talking as long as they wished on any topic they wished (see the Jimmy Stewart movie Mr. Smith Goes to Washington) until cloture (closure of debate) is called. Cloture, however, requires the votes of three fifths of the senators. In today's Senate 60 senators would have to vote to end the filibuster.
Moreover, filibusters today are not carried out by interminable speeches but are simply declared, and until one party can muster 60 votes for cloture the measure being filibustered is essentially dead.
Not everything the Senate does is subject to the filibuster, but legislation often is so those bills essentially require a 60 vote majority in order to be enacted into law.
Since this is merely a rule of the Senate and not required by the Constitution - even though it's been around in one form or another since the early 1800s - it could be changed by a simple majority, and that's what many Democrats want to do in order to prevent the Republicans from using the filibuster to block their agenda.
The Democrats have a slender majority even though the Senate is split 50/50 between Democrats and Republicans because the vice-president can break a tie. Presently, absent a filibuster, any tie would be broken by Kamala Harris in favor of the Democrats.
Rachel Bovard at The Federalist explains why the filibuster is important:
While the filibuster is often decried for slowing down, hamstringing, or blocking the legislative process, it was actually designed as a way to facilitate it. Generally speaking, passing legislation in the Senate requires the participation of more than just the majority.Since it would take fifty senators plus Vice-president Harris to approve of doing away with the filibuster, if all the Republicans wanted to keep it it appears right now that it would survive since at least two Democrats, Kyrsten Sinema of Arizona and Joe Manchin of West Virginia, have said they don't support its abolition.
Ending a filibuster — that is, getting to 60 votes in the Senate — requires consensus. It forces the parties to work together, to engage in negotiation, and to perform the give-and-take of legislating. This deliberation is what distinguishes the Senate from the House, where the majority has the full authority to crush the minority, and frequently does.
Furthermore, the filibuster performs an additional important function: it gives individual senators — and the causes that lack support of the majority — a voice they otherwise wouldn’t have. Because overcoming a filibuster requires a large consensus, causes can’t simply be steamrolled.
The concerns of individual senators, or groups of senators, must be given both credence and credibility. Voices and causes that would otherwise be ignored in a majoritarian body like the House receive consideration in the Senate — but only because the filibuster, or threat thereof, makes them matter.
The filibuster, in other words, amplifies otherwise voiceless causes and makes certain that they are taken seriously. For conservatives, as well as the far left of the Democratic Party, who are generally always in the minority even when their party is in the majority, the filibuster is a powerful tool.
