A pivot is a chord which belongs to both the old and new keys. Pivot chords can make modulations smoother, but it is not essential to use a pivot chord in a modulation.
This tune starts in C major, then modulates to the dominant key of G major, via a pivot chord of A minor in bar 5. Pivot chords are identified by writing the Roman numeral of the old key = numeral of the new key, e.g. “vi=ii” in this example.
Pivot chords allow you to use chords which might not otherwise be typical in a progression. For example, in C major chord iii is only rarely used, but at a modulation to the dominant it can be used as a iii=vi pivot chord to good effect, as it is here.
(NB this chord is analysed as E minor (iii) and not C major (I), because the leading note in the previous bar would have risen to C, rather than falling to E, in a V-I progression).
We have already seen this example of a modulation without a pivot chord (in the section on Modulation). In this modulation to the subdominant, the chord before V7 in F major is G major, which does not belong to the new key: