Sometimes a song does not begin on the first beat of a bar. Any notes which come before the first full bar are called lead-in notes (or pick-up notes). When lead-in notes are used, the last bar is also incomplete. The notes in the lead-in and the last bar add up to one full bar.
When the Saints Go Marchin’ In is an early Jazz standard made popular by brass bands in New Orleans. The song uses a lead-in and also contains both quarter and half rests. The counting numbers refer to the melody (right hand part). Instead of writing a chord symbol above each bar of music, it is common to only to write a chord symbol when the chord changes, e.g., the first 6 bars of this song are a C chord.