After a bill has been sent to the president it becomes a law if he does not send it back?

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