The president signs a bill after the legisilative branch approves it. The president (executive branch) enforces or carries out a law ( or bill. )
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When the president signs the bill, it becomes law. If the president refuses, the bill is vetoed, but if a two thirds vote by Congress, it can still become law.
In the United States, when a bill that has been approved by both Houses of Congress, it is sent to the President. Thus a bill can become a law when the President signs it into law.
The bill doesn't become law unless the president signs it. When the president receives a bill from Congress he has two options 1) he may veto it; which is to essentially reject it or 2) he can sign it; it then becomes law. If a president ignores a bill that is passed by Congress for 10 days, it passes with or without his signature. There is an exception, the "pocket veto." If a president ignores a bill and Congress adjourns, the bill dies (as if he folded it up and stuck it in his pocket). http://en.wikipedia.org/wiki/Pocket_veto
At this point, the bill is sent to the President. If he or she signs it, the bill becomes law. If the President vetoes the bill, it is sent back to congress. Now, it will take a 2/3 vote in both the Senate and House of Representatives to override the President's veto and make the bill a law. (It is very hard to get a 2/3 vote.) There is also a pocket veto where the President does nothing and just lets the bill die. (Look up pocket veto.)
The bill automatically dies. For a president to allow that to happen is known as a pocket veto.