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 a 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 use a C chord.