What does "f" stand for in C standard library function names?

The leading f refers to the type that function operates on:

• fgets: use gets on a FILE * handle instead of just stdin
• fopen: open a file, and return it as a FILE * (instead of a file descriptor which the original open does)

The trailing f means that it uses a formatting string:

• printf: print out according to the format specifier
• scanf: read in according to the format

And combined, you get things like:

• fprintf: print out to a particular FILE * according to the format specifier

When you consider things like the math.h functions, then the trailing f designates that the particular function operates on operands of type float like so:

• powf: take the exponent of floats
• powl: take the exponent of long doubles

Your question in general is too general but I can explain a few examples.

• fgets, fopen, fclose, … — The ”f“ stands for “file”. These functions accept or return a FILE * pointer as opposed to a file number as the POSIX functions do.
• printf, scanf, … — The ”f“ stands for “formatted”. These functions accept a format string.
• fprintf, fscanf — This is a combination of the above two.
• sinf, cosf, … — The “f” stands for float (to distinguish from the double alternatives). Note that this fits quite nicely with suffixing floating point literals with an f as in 1.5f.
• Finally, as Deduplicator points out, there are some names such as free, floor or setbuf (“set buffer”) where the “f” simply appears as a natural language character.

The tradition of pre- or suffixing names with single letters that indicate the type of the arguments is a necessity in C that has become obsolete in C++ thanks to overloading. Actually, overloading in C++ works by the compiler automatically adding those suffixes again under the hood to the generated symbols by a process called name mangling.