Q:

What does it mean to have an exception for no mode? (The number that occurs the most in an array)?

When the problem is executed it works correctly stating that the mode is 3. The professor asked for "The program should account for the “exception” that no “mode” occurs – no value in the array occurs more than once." The thing is I do not understand what it is being asked. I cannot wrap my head around starting such a thing. 
void showArray(const int[], int); 
void showMode(int [], int);

using namespace std;

int main()
{
    const int Size = 11; //size of the array
    int test[Size] = {1, 2, 3, 3, 3, 2, 2, 1, 3, 4, 5}; //elements of the array

    cout << "The Numbers in the Array are: \n";
    showArray(test, Size); //displays the array in its original order

    showMode(test, Size);

    return 0;
}

void showArray(const int arr[], int Size) 
{
    for(int count = 0; count < Size; count++)
        cout << arr[count] << " ";
    cout << endl;

}

void showMode(int test[], int Size)
{
    int counter = 1;
    int max = 0;
    int mode = test[0];

    for(int pass = 0; pass < Size - 1; pass++)
    {
        if(test[pass] == test[pass + 1])

            {
                counter++;
                if(counter > max)
                {
                    max = counter;
                    mode = test[pass];
                }
            }
        else
            counter = 1;
    }
    cout << "The Mode of the Array is: " << mode << endl;
}

A:

When testing software, you want to have test cases that take all the different branches.  Right now you've only demonstrated that the code works when there is a unique highest-frequency element for the given input array.
Your professor also wants you to test when the highest repetition count is not unique.
This is often called a corner case.
The professor's phrasing is unfortunate.  The word exception has a specific meaning in C++, and this is not it... unless your instructions are to actually throw an exception when the highest repetition count isn't unique.
Good additional test cases would be:

Array of length zero
{}

Array of length one
{ 7 }

2-way tie for highest repetition count
{ 1, 2, 3, 2, 1 }
{ 1, 1, 2, 3, 3 }

N-way tie for highest repetition count
{ 1, 3, 5, 4, 2 }

Tie is broken by first element
{ 2, 6, 3, 2, 4, 5, 2, 6, 9 }

Tie is broken by last element
{ 6, 3, 2, 4, 5, 2, 6, 2 }

Other element has longer "run"
{ 5, 2, 6, 3, 2, 4, 4, 5, 2, 6, 9 }

If your code gives right results for all these cases, you will have high confidence that it is right for all inputs.

