The C we know as of today is ANSI C, which is the standardized version of the C that Dennis Ritchie had developed during 1969–73. Sometimes it is said that the 'C' stands for Compiler. C+ was till 1983 'C with Classes' (as named by Bjarne Stroustrup, its developer).

Seem to be a lot of answers here implying C then C+ was the culmination of a lot of things. Actually, if you follow it through the successors just got worse and worse.

The start of structured programming (see the book Structured Programming by Dahl, Dijkstra, and Hoare) was ALGOL (before that the start of high-level programming was FORTRAN and COBOL, but ALGOL was the first really decent language, out of which most else comes).

The ideas were thick and fast for ALGOL and it spawned off many new tracks. The one here was CPL (from London and Cambridge) Combined Programming Language. This might have been an improvement on ALGOL, but it was then cobbled by BCPL (Basic-CPL), then made worse by B and then C. Then the granddaddy of complete obfuscation and bad design C+, which picked up on the variant track of ALGOL, Simula which started OO (also described in Structured Programming). But C+ completely messed up these ideas.

As Tony Hoare once quipped “ALGOL was a significant advance on most of its successors”!

So really the “full form” of C is ALGOL and C+ is Simula and languages like Smalltalk and Eiffel.