What is the effect of extern “C” in C++?

extern "C" makes a function-name in C++ have C linkage (compiler does not mangle the name) so that client C code can link to (use) your function using a C compatible header file that contains just the declaration of your function. Your function definition is contained in a binary format (that was compiled by your C++ compiler) that the client C linker will then link to using the C name.

Since C++ has overloading of function names and C does not, the C++ compiler cannot just use the function name as a unique id to link to, so it mangles the name by adding information about the arguments. A C compiler does not need to mangle the name since you can not overload function names in C. When you state that a function has extern "C" linkage in C++, the C++ compiler does not add argument/parameter type information to the name used for linkage.

Just so you know, you can specify extern "C" linkage to each individual declaration/definition explicitly or use a block to group a sequence of declarations/definitions to have a certain linkage:

extern "C" void foo(int);
extern "C"
{
   void g(char);
   int i;
}

If you care about the technicalities, they are listed in section 7.5 of the C++03 standard, here is a brief summary (with emphasis on extern "C"):

extern "C" is a linkage-specification
Every compiler is required to provide "C" linkage
A linkage specification shall occur only in namespace scope
All function types, function names and variable names have a language linkage See Richard's Comment: Only function names and variable names with external linkage have a language linkage
Two function types with distinct language linkages are distinct types even if otherwise identical
Linkage specs nest, inner one determines the final linkage
extern "C" is ignored for class members
At most one function with a particular name can have "C" linkage (regardless of namespace)
extern "C" forces a function to have external linkage (cannot make it static) See Richard's comment: static inside extern "C" is valid; an entity so declared has internal linkage, and so does not have a language linkage
Linkage from C++ to objects defined in other languages and to objects defined in C++ from other languages is implementation-defined and language-dependent. Only where the object layout strategies of two language implementations are similar enough can such linkage be achieved

Just wanted to add a bit of info, since I haven't seen it posted yet.

You'll very often see code in C headers like so:

#ifdef __cplusplus
extern "C" {
#endif

// all of your legacy C code here

#ifdef __cplusplus
}
#endif

What this accomplishes is that it allows you to use that C header file with your C++ code, because the macro "__cplusplus" will be defined. But you can also still use it with your legacy C code, where the macro is NOT defined, so it won't see the uniquely C++ construct.

Although, I have also seen C++ code such as:

extern "C" {
#include "legacy_C_header.h"
}

which I imagine accomplishes much the same thing.

Not sure which way is better, but I have seen both.