This line:

m_pchString[m_nLength-1] = '';

What you probably mean is:

m_pchString[m_nLength-1] = '\0';

Or even:

m_pchString[m_nLength-1] = 0;

Strings are zero terminated, which is written as a plain 0 or the null character '\0'. For double quote strings "" the zero termintation character is implicitly added to the end, but since you explicitly set a single character you must specify which.

The empty character constant error is because of the following comparison in the else if clause.

c == '' //  '' is empty

Replace '' with ' '.

You can use c[i]= '\0' or simply c[i] = (char) 0.

The null/empty char is simply a value of zero, but can also be represented as a character with an escaped zero.

Can we put it in C++ source code?

No, it would be a syntax error.

If not, what's the usage of ''?

There is no usage, unless your purpose is to cause a compilation error (for which there are probably better alternatives such as static_assert).

Can it be understood that empty character constant '' is just a pure grammar error just like a variable being named as 2018ch ?

Yes. The grammar says:

character-literal:
     encoding-prefix opt ' c-char-sequence '

Notice that unlike the encoding-prefix, c-char-sequence is not optional.

Side note: Yes, it is a character sequence - multi character literals exist. But you don't need to learn about them yet other than knowing that you probably won't need them. Just don't assume that they're strings.

No, in order to remove a character in a string you will have to move the rest of the string one step, you cannot simple replace it with "empty" character. You could use the erase method though, but then you should probably not do that while iterating the string.

What you probably should do is to build a new string as you traverse the original string, something like:

string rem_vow(string const& s)
{
    string res;

    for (char c : s)
    {
        switch (c)
        {
        case 'A': case 'a': case 'E': case 'e': case 'I':
        case 'i': case 'O': case 'o': case 'U': case 'u':
            //c = ' ';
            break;
        default:
            res.push_back(c);
            break;
        }
    }

    return res;
}