The C major chord is constructed from the C major scale. Using the chord formula 1 3 5 on the C major scale, it can be seen that the C major chord contains the notes C, E and G.
Here is a C chord in the bass staff, along with a keyboard diagram demonstrating the notes.
Look again at the first two bars of Banks of the Ohio (Version 2) and notice that all the notes in both the melody and harmony are contained within the C chord. Notice also the C chord symbol telling the band or accompanist that this is the right chord to play at this point in the song.