So everything works right now, but I need that final section (will be turned into a function later) to display the number of A's, B's, C's, D's, and F's. It seems like it should be pretty simple, but nothing is working for me. Thanks!

#include <iostream>

using std::cout;
using std::cin;
using std::endl;

int main()
{
    int scores[10],
        average = 0,
        grade [10];
    double score = 0;

    for ( int i = 0; i < 10; i++ )
    {
        cout << "Enter Student #" << i + 1 << "'s score: ";
        cin >> scores[i];

        score = scores[i];

     if ( score >= 92.0 ) 
        cout << "You scored a " << scores[i] << ". You received an A" <<'\n';
        else
            if ( score >= 84.0 && score < 92.0 )
                cout << "You scored a " << scores[i] << ". You received a B" <<'\n';
            else
                if ( score >= 75.0 && score < 84.0 )
                    cout << "You scored a " << scores[i] << ". You received a C" <<'\n';
                else
                    if ( score >= 65.0 && score < 75.0 )
                        cout << "You scored a " << scores[i] << ". You received a D" <<'\n';
                    else
                        if ( score < 65.0 )
                            cout << "You scored a " << scores[i] << ". You received an F" << '\n'; 
    }

    average = ( scores[0] + scores[1] + scores[2] + scores[3] + scores[4] + scores[5] + scores[6] + scores[7] + scores[8] + scores[9] ) / 10;

    cout << "Average class score: " << average <<'\n';

    cout << "Number of A's: " <<'\n'
         << "Number of B's: " <<'\n'
         << "Number of C's: " <<'\n'
         << "Number of D's: " <<'\n'
         << "Number of F's: " <<'\n';

    return 0;
}

Keep a set of counter variables to count the number of A's, B's, C's etc.
Initialise them to zero at the beginning of your program, increment the appropriate counter when a grade is assigned in your block of if..else if statements.

e.g.
Declare some counters with the rest of your variables:

int numAs, numBs, numCs, numDs, numFs;
numAs = numBs = numCs = numDs = numFs = 0;

Increment the appropriate counter in your if..else if block:

if ( score >= 92.0 ) 
{
    cout << "You scored a " << scores[i] << ". You received an A" <<'\n';
    ++numAs;
}
else if ... // I'll leave you to do the rest!

Finally in your output section:
cout << "Number of A's: " << numAs << '\n'
Again, I'll leave you to fill in the rest of the blanks!
:)