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Is it true that the first C compiler was written in C itself? Then, how was it executed and compiled? Or, was this compiler written in assembly language?
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To directly answer your question: the first compiler was written (by a human) in an assembly language -- a program called an assembler would translate assembly language into binary; this is a much simpler process than compilation because assembly language is just a symbolic form of machine language that uses opcode ...
C was first written in B which was a stripped down BCPL which was a stripped down Algol 60 I believe. B was probably written in a macro assembler though.
The first C compiler, written by Dennis Ritchie, used a recursive descent parser, incorporated specific knowledge about the PDP-11, and relied on an optional machine-specific optimizer to improve the assembly language code it generated.
A very simple compiler can be written from an assembler and machine code. Once you have a software that is able to translate something into binary instructions, you can use the original compiler to write a more sophisticated one (then use a second further refined one to write a third and so on).
It is described pretty well in Dennis Ritchie's writeup of the C language history.
Giving just a summary of what he wrote there, use his article for the finer details. C started with the BCPL language, Ken Thomson had access to a compiler for it that ran on their General Electrics 635 main frame. Unhappy with the language, Thomson used BCPL to write a compiler for the B language, an evolutionary step beyond BCPL that removed some of the technical problems in BCPL.
They used B to create programs on their PDP-7 minicomputer, although most of them were little more than toy programs, the machine had very restricted hardware capabilities. One important step taken by Thomson was to rewrite the B compiler in B itself. A common bootstrapping step.
That compiler was then gradually tweaked, through a short-lived intermediary step called NB (New B) to start resembling C. The PDP-11 minicomputer was instrumental in that step, giving them enough room to improve the language and compiler.
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The first C compiler wasn't written in C, usually when writing a compiler we use either assembly language, or another programming language, and it's common that after first compilation, the compiler is rewritten in it's native language.
There's a lot of programming languages that have been written in C then rewritten in their native language : Java for example, Ada ...
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