The American Immigration Lawyers Association ("AILA") explains what is happening as S.744 is debated on the Senate Floor:

Process (Starting the Week of June 10th)  

Senate Majority Leader, Harry Reid (D-NV), filed cloture on the motion to proceed on 6/6/13

 The Senate will debate the motion to proceed for two calendar days and will conclude with a cloture vote on 6/11/13 at 2:15pm and a final vote at 4:00pm.  If cloture is

invoked (requires 60 yes votes—which should not be a problem as Minority Leader McConnell (KY) has indicated he will vote yes), and the motion to proceed passes then the Senate will begin debate of the bill.

 The Congressional Budget Office (CBO) will release a score for the bill—estimating how much the bill will cost the government (add or reduce the deficit) over the next ten years.

 Debate will begin with opening statements from both parties and then time will be divided evenly between Democrats and Republicans

o Sen. Leahy (D-VT), Chairman of the Judiciary Committee, will work with Sen. Reid and the four Democratic members of the “Gang of Eight” to lead the Democratic efforts for the bill.

o For Republicans, the time will be divided between proponents (the four Republican members of the “Gang of Eight” and others) and opponents (Senators Sessions (AL), Grassley (IA), and others).

 Amendments (offered over the next three weeks) see attached  filed as of 6/11/2013 http://www.immigrationbn.com/Portals/52046/docs/amend.pdf

 We anticipate that hundreds of amendments will be filed to S. 744; however, just because an amendment is filed does not mean it will be called up for a vote—in fact most won’t (similar to what happened in the committee process).  

 Time will be allotted to debate the amendments. Many (typically non-controversial) amendments will be accepted on a bipartisan basis.  We expect that the most controversial amendments will be held off until the third week of debate (June 24th), right before the cloture vote on the bill.

 Some amendments may require a higher 60 vote threshold, while others will only need a simple majority.

 Senators may offer amendments in any order to any part of the bill that has not already been amended. The order in which amendments are offered depends largely on the convenience of the Senators proposing them, not on requirements imposed by standing rules or precedents. 

 Senators can offer 2nd degree amendments to any amendment being debated on the bill.  After voting on any second degree amendments, the Senate votes on the first degree amendment as it may have been amended. Third degree amendments—amendments to second degree amendments—are not in order.

Final Votes on Bill as Amended (Most Likely Week of June 24th)

 Filibuster: Because Senate rules establish no generally applicable limits on the length of debate, nor any motions by which a majority could vote to bring a debate to an end, or even limit it, the only formal procedure that Senate rules provide for breaking filibusters (endless amendments being offered, one senator speaking at length, etc…) is to invoke cloture.  

 When Senator Reid has decided that the time for offering amendments has expired, he will invoke cloture to end debate on the bill.  If any Senator objects to cloture it will then require 60 yes votes to end debate and move to a final vote.  If cloture is invoked successfully (60 yes votes) there will still be an additional 30 hours (or two calendar days) of debate when amendments can continue to be offered.

 If cloture is invoked, and after the bill has ripened (the 30 hours have elapsed), the Senate will then move to a final vote on the bill as amended.  This vote only requires a simple majority of the Senate for passage.