So today I figured (for the first time admittedly) that int foo() is in fact different from int foo(void) in that the first one allows any number of inputs and the second one allows zero.
Does int foo() simply ignore any given inputs? If so, what’s the point of allowing this form of function? If not, how can you access them and how is this different from having a variable argument list (e.g. something like int foo (int argc, ...))?
The key to understanding this is understanding the call stack. What happens when you call a function in C (with the standard x86 ABI — other platforms may vary) is that the caller pushes all of the arguments, in reverse order, onto the stack before calling the callee. Then, the callee can read as many of those arguments as it wants. If you use foo(void), obviously you won’t read any. If you use foo(int), you’ll be able to read one word into the stack below your current frame.
Using foo() (with no args specified) means that the compiler won’t care to check the arguments you pass to foo. Anything goes; there’s no contract. This can be useful if foo has two different implementations that take different types as arguments, and I want to be able to switch them out programmatically. (It’s also sometimes useful when writing functions in assembler, where you can handle the stack and arguments yourself.) But remember that the compiler is not going to do any type-checking for you, so you have to be very careful.
This is different from foo(int, …), since in that case, the function actually has a way to access all of the arguments (using varargs). There’s no reason ever to actually define a function with foo(), since it’s basically equivalent to foo(void). foo() is only useful for declarations.