An arpeggio is a chord played one note at a time. The value of arpeggios is that they enable you to play parts which fit chord progressions perfectly, since every note of an arpeggio is a note of the accompanying chord.
Written below is a C major arpeggio which consists of the notes C, E and G. These are the root, third and fifth of a C major chord. To play this arpeggio, simply hold down a C chord shape and pick the notes individually.
Here is an example which makes use of arpeggios. Hold down the chord shapes indicated by the chord symbols as much as possible, except when playing notes which are not in the chords (e.g. the B note at the end of bar 2). Take care with your picking and make sure all your notes are clear and even.