The C major chord is constructed from the C major scale. Using the major chord formula on the following C major scale, it can be seen that the C major chord contains the notes C, E and G.
The chord symbol for a major chord is just the letter name of the chord, E.g., the chord symbol for the C major chord is C. It is common practice to refer to a C major chord as the C chord. This abbreviation applies to all major chords.
Play the three notes of the C chord with the first, third and fifth fingers of your right hand, individually, and then together as shown here.
When playing a chord one note at a time it is referred to as an arpeggio.
The first note from the major scale (and hence the first note of the major chord) is called the root note. When playing chords with the right hand, the left hand often plays the root note, as shown in example 1.
In example 1 the C chord is played in arpeggio style in bars 1 and 3.