Re: Actual question
The executive branch (under the president) negotiates foreign treaties and the president signs them.
Then to come into effect properly the foreign treaty has to be ratified by the Senate.
https://en.wikipedia.org/wiki/List_of_United_States_treaties
"Under the treaty clause of the United States Constitution, treaties come into effect upon final ratification by the President of the United States, provided that a two-thirds majority of the United States Senate concurs."
Note that the USA often makes use of "Reservations, understandings, and declarations". on treaties, things that state the USA's interpretation of the treaty. If you go looking you'll often see these clauses say things like, "The United States of American understands this treaty as not in any way restricting its ability to [do what it has always done/do whatever it wants/carry on as before]".
I used UN Treaties for examples, because they can be found easily on the web and they're usually easier for us laypeople to understand than trade treaties.
For example:
http://hrlibrary.umn.edu/usdocs/civilres.html
which makes the USA's signature on this treaty arguably meaningless within the USA
http://www.ohchr.org/en/professionalinterest/pages/ccpr.aspx
Another example:
https://treaties.un.org/Pages/ViewDetails.aspx?src=TREATY&mtdsg_no=IV-9&chapter=4&clang=_en#13
for this treaty:
http://www.ohchr.org/EN/ProfessionalInterest/Pages/CAT.aspx