Q: Is C++ a superset of C? What are the differences between C and C++? Can I use a C++ compiler to compile C code?
A: C++ was derived from C, and is largely based on it, but there are some legal C constructs which are not legal C++. Conversely, ANSI C inherited several features from C++, including prototypes and const, so neither language is really a subset or superset of the other; the two also define the meaning of some common constructs differently.
The most important feature of C++ not found in C is of course the extended structure known as a class which along with operator overloading makes object-oriented programming convenient. There are several other differences and new features: variables may be declared anywhere in a block; const variables may be true compile-time constants; structure tags are implicitly typedeffed; an & in a parameter declaration requests pass by reference; and the new and delete operators, along with per-object constructors and destructors, simplify dynamic data structure management. There are a host of mechanisms tied up with classes and object-oriented programming: inheritance, friends, virtual functions, templates, etc. (This list of C++ features is not intended to be complete; C++ programmers will notice many omissions.)
Some features of C which keep it from being a strict subset of C++ (that is, which keep C programs from necessarily being acceptable to C++ compilers) are that main may be called recursively, character constants are of type int, prototypes are not required, and void * implicitly converts to other pointer types. Also, every keyword in C++ which is not a keyword in C is available in C as an identifier; C programs which use words like class and friend as ordinary identifiers will be rejected by C++ compilers.
In spite of the differences, many C programs will compile correctly in a C++ environment, and many recent compilers offer both C and C++ compilation modes. (But it's usually a bad idea to compile straight C code as if it were C++; the languages are different enough that you'll generally get poor results.)
See also questions 8.9 and 20.20.
Bjarne Stroustrup's answer on the subset/superset question
an article by Richard Stamp listing some differences
an article by ``Noone Really'' listing some more
H&S p. xviii, Sec. 1.1.5 p. 6, Sec. 2.8 pp. 36-7, Sec. 4.9 pp. 104-107