What is the general theory that describes the interactions between strings? I mean the basic object in the theory is (closed) string and they have interactions among them. The string theory, as I learned now (I am far from an expert!), describes only one string. Can anyone recommend a reference of such a theory?
String theory is the theory of the propagation and interaction of strings. Any textbook on String theory will have several chapters devoted to the calculation of string scattering amplitudes.
string theory does not describe only one string, quite the opposite, each particle is a string. The theory also describe the interactions between particles, and so, between strings.
As other answers have pointed out already, String theory describes not only one string, the basic objects are open and closed strings, also there are some other fundamental ones such as D-branes. As a firrst book to read I would recommend Zwiebach, "A first course in String Theory".