The name "C" refers to a family of related languages, some of which are formalized as international standards. These include K&R C, ANSI C, C99, and C11.
The name "C++" refers to a family of related languages, some of which are formalized as international standards. These include
C++98,
C++03,
C++11,
C++14,
and the speculated C++17.
The term "C/C++" is used by many people to informally refer to the intersection of C and C++, which has been intentionally maintained by the designers of C++.
C++'s immediate predecessor was in fact originally called "C with Classes". This is detailed in Bjarne Stroustrup's 1994 book "The Design and Evolution of C++" (and also here). The name was eventually changed to "C++", largely as a courtesy to the C community because it had become too tempting for people to shorten "C with Classes" to just "C" or "new C".
There are many language features in C++ that are not in C (the reverse is true to a lesser extent). In particular, the class-based model, due to its capability to invoke implicit function calls, is so powerful that C++ has (rather organically) evolved higher-level programming styles that make the typical usage of C++ much different than that of C. The most notable examples of this are associated with the concept called Resource Acquisition Is Initialization, which is embodied in the standard classes
string, vector, and shared_ptr as well as the stream classes of the standard I/O library.
Other notes:
GCC allows you to specify which of these language variations you want to compile against (as do other compilers, I'm sure), with some limitations.
C++98 and C++03 have minimal feature differences and are often treated as essentially the same language.
Top questions that address the common subset of C and C++: