A bill is introduced in either chamber (except taxes and appropriations, which must originate in the House). It gets referred to a committee (by the Speaker in the House, and (I think) the Senate Majority Leader in the Senate). That committee holds hearings and considers amendments, if it considers the bill a viable candidate for passage (if not, the bill is allowed to die in committee). The committee then reports its version of the legislation (or, in very rare instances, it can be discharged from the committee by the body that sent it to committee–either the House or the Senate).
It gets read a couple of times, possibly considered in the Committee of the Whole (a committee consisting of every member of the House; I’m not certain if this exists in the Senate and if it’s called the same thing), amendments are taken from the floor, etc. etc. The House Rules Committee will set the parameters of debate in the House of Representatives.
Finally, a vote is taken. Most legislation, in either chamber, requires only a simple majority, that is, one-half of those voting plus one(with a least a quorum present). If it passes, it is sent to the other chamber for consideration. Recall that because a bill can originate in either chamber (with exceptions like that noted above), it might thus be sent to the Senate or to the House.
That other chamber then considers it, much like as in the first. Then they vote. And again, usually only a simple majority is required (sixty is needed for a petition of cloture in the Senate, which is the device used to terminate debate/filibusters and put the bill to a vote). If it passes and the bills are identical–Great! It gets sent to the President. If not, a conference committee composed of members from the House and Senate is formed and they attempt to work out a compromise version that is sent back to both chambers for passage. At this time, the conference report (the name of the proposed compromise legislation) is not open to amendment–for obvious reasons.
If this conference report is passed by both, it is then sent to the President for signature (or veto). If the President signs it, the bill becomes a law. A veto is accomplished when the bill is returned, with ten days, exclusive of Sundays, after its passage in Congress, unsigned to the chamber in which it originated together with the President’s objections to the legislation. The Constitution requires this statement of objections, and if the President does not sign, but also does not object, the bill becomes law after the ten days, exclusive of Sundays, elapses. If it is vetoed, the Congress may override it by repassing the legislation, with a two-thirds majority in each chamber (note the higher passage bar on the second time around). Unless either chamber adjourns sine die (which, as a side note, requires the consent of the other chamber) before that deadline passes, which effectively vetoes the legislation. This latter process is called the “pocket veto.”