Comparing 2 Strings, one in a struct other not C programming

int checkProduct (char nameCheck[100])

Note that the type signature is a lie. The signature should be

int checkProduct(char *nameCheck)

since the argument the function expects and receives is a pointer to a char, or, to document it for the user that the argument should be a pointer to the first element of a 0-terminated char array

int checkProduct(char nameCheck[])

Arrays are never passed as arguments to functions, as function arguments, and in most circumstances [the exceptions are when the array is the operand of sizeof, _Alignof or the address operator &] are converted to pointers to the first element.

    {
        product temp;
        p.pName = nameCheck;

Arrays are not assignable. The only time you can have an array name on the left of a = is initialisation at the point where the array is declared.

You probably want

strcpy(p.pName, nameCheck);

there.

        rewind (pfp);
        while (fread(&temp,STRUCTSIZE,1,pfp)==1)
        {
            if (strcmp (temp.pName,p.pName))

strcmp returns a negative value if the first argument is lexicographically smaller than the second, 0 if both arguments are equal, and a positive value if the first is lexicographically larger than the second.

You probably want

if (strcmp(temp.pName, p.pName) == 0)

there.

    gets (nameCheck);

Never use gets. It is extremely unsafe (and has been remoed from the language in the last standard, yay). Use

fgets(nameCheck, sizeof nameCheck, stdin);

but that stores the newline in the buffer if there is enough space, so you have to overwrite that with 0 if present.

If you are on a POSIX system and don't need to care about portability, you can use getline() to read in a line without storing the trailing newline.

    checkProduct (nameCheck);

You check whether the product is known, but throw away the result. Store it in a variable.

    while (checkProduct == 1)

checkProduct is a function. In almost all circumstances, a function designator is converted into a pointer, hence the warning about the comparison between a pointer and an integer. You meant to compare to the value of the call you should have stored above.

    {
        printf ("Product Already Exists!\n Enter another!\n");
        while (getchar() !='\n')

You read in characters without storing them. So you will never change the contents of nameCheck, and then be trapped in an infinite loop.

        {
            continue;
        }

If the only statement in a loop body is continue;, you should leave the body empty.

    }
    p.pName = nameCheck;

Once again, you can't assign to an array.

Concerning the edit,

char *nameCheck;
nameCheck = "";
fgets(nameCheck,sizeof nameCheck, stdin);

you have changed nameCheck from an array to a pointer. That means that sizeof nameCheck now doesn't give the number of chars you can store in the array, but the size of a pointer to char, which is independent of what it points to (usually 4 on 32-bit systems and 8 on 64-bit systems).

And you let that pointer point to a string literal "", which is the reason for the crash. Attempting to modify string literals is undefined behaviour, and more often than not leads to a crash, since string literals are usually stored in a read-only segment of the memory nowadays.

You should have left it at

char nameCheck[100];
fgets(nameCheck, sizeof nameCheck, stdin);

and then you can use sizeof nameCheck to tell fgets how many characters it may read, or, alternatively, you could have a pointer and malloc some memory,

#define NAME_LENGTH 100
char *nameCheck = malloc(NAME_LENGTH);
if (nameCheck == NULL) {
    // malloc failed, handle it if possible, or
    exit(EXIT_FAILURE);
}
fgets(nameCheck, NAME_LENGTH, stdin);

Either way, after getting input, remove the newline if there is one:

size_t len = strlen(nameCheck);
if (len > 0 && nameCheck[len-1] == '\n') {
    nameCheck[len-1] = 0;
}
// Does windows also add a '\r' when reading from stdin?
if (len > 1 && nameCheck[len-2] == '\r') {
    nameCheck[len-2] = 0;
}