std::string s = std::string("foo");
This creates a temporary std::string object containing “foo”, then assigns it to s. (Note that compilers may elide the temporary. The temporary elison in this case is explicitly allowed by the C++ standard.)
std::string s = new std::string("foo");
This is a compiler error. The expression new std::string("foo") creates an std::string on the free store and returns a pointer to an std::string. It then attempts to assign the returned pointer of type std::string* to s of type std::string. The design of the std::string class prevents that from happening, so the compile fails.
C++ is not Java. This is not how objects are typically created, because if you forget to delete the returned std::string object you will leak memory. One of the main benefits of using std::string is that it manages the underlying string buffer for you automatically, so new-ing it kind of defeats that purpose.
std::string s = "foo";
This is essentially the same as #1. It technically initializes a new temporary string which will contain “foo”, then assigns it to s. Again, compilers will typically elide the temporary (and in fact pretty much all non-stupid compilers nowadays do in fact eliminate the temporary), so in practice it simply constructs a new object called s in place.
Specifically it invokes a converting constructor in std::string that accepts a const char* argument. In the above code, the converting constructor is required to be non-explicit, otherwise it’s a compiler error. The converting constructor is in fact non-explicit for std::strings, so the above does compile.
This is how std::strings are typically initialized. When s goes out of scope, the s object will be destroyed along with the underlying string buffer. Note that the following has the same effect (and is another typical way std::strings are initialized), in the sense that it also produces an object called s containing “foo”.
std::string s("foo");
However, there’s a subtle difference between std::string s = "foo"; and std::string s("foo");, one of them being that the converting constructor can be either explicit or non-explicit in the above case.