As with most official meetings, both the House and Senate operate under traditional parliamentary rules, or Robert's Rules of Order. Parliamentary rules date back to Greek government and keep the agenda organized, the proceedings moving, and the debate civil (at least theoretically). Parliamentary rules require a call to order, a formal motion to propose any action, the seconding (or approval) of any motion, the keeping of "minutes" as the official record of every meeting, and many other guidelines. You'll see Roberts Rules in place if you watch Congress in session or attend most any city council, school board, or board of directors meeting.
Any member of the House can introduce a bill at any time during a session. Once introduced, the Speaker of the House assigns the bill to a committee (for example, the Science and Technology committee may review a bill on climate change). The committee closely examines facts and existing laws and may conduct hearings of experts and other parties who have an interest in the bill. The committee may also amend (or change) the bill, and then votes to either reject the bill or send it to the full House.
Once a bill is sent to the full House, it gets added to the agenda (or House Calendar) and gets a designated time allowance for debate. Generally, the supporters and opponents of a bill control debate time (often this is the Chairman and Ranking member of the bill's primary committee), but any representative may yield time to other representatives who wish to speak about the bill. The bill may be amended further, and after the conclusion of time for debate and all amendments have been addressed, the bill is voted upon by the full House — unless a representative rules to send it back to committee.
If the bill passes the in the House, it moves on to the Senate. Unlike debates in the House, Senate debates do not have a time constraint, but senators may only speak if called upon by the Presiding Officer. The presiding officer has little control over the debate, however, because he or she is required to recognize the first senator who rises, unless either the Majority or Minority Leader wishes to speak. The two leaders trump other senators.
There aren't any restrictions on what senators can speak about; they don't even have to stay on the topic of the bill being debated. But senators are prohibited from making more than two speeches on a bill on the same legislative day.
A filibuster is a tactic used to defeat bills in the Senate by prolonging debate indefinitely. A filibuster may include very long speeches, a lengthy series of proposed amendments, or a few other stalling tactics. The Senate may end a filibuster by invoking cloture, which usually requires approval by three-fifths of the Senate. Cloture is therefore invoked rarely, because to get that many senators on board requires bipartisan support. Note there is no equivalent to the filibuster in the House.
When debate concludes, the bill is put to a vote. If a senate vote is tied, the Vice President of the United States, if present, is entitled to vote. (If the vice president is not present, the motion fails.)
Once the bill passes both houses of Congress, it goes to the president to sign.