One of the most common forms of playing off the beat in piano music is a rhythmic concept called syncopation. To understand syncopation, you have to know about downbeats and upbeats. Start tapping your foot to a moderate 4/4 beat, and count eighth notes “1-and, 2-and, 3-and, 4-and.” Your foot goes down on the downbeats and up on the upbeats.
Downbeats are normally emphasized in a song. But through the miracles of syncopation, you emphasize some (or all) of the upbeats instead. So you play those notes a little bit harder, or louder, than the others. For example, the note normally played on beat 3 is played on the upbeat before, and this anticipation naturally emphasizes the upbeat and creates syncopation.
A two-measure melodic phrase written first to emphasize the downbeat on beat 3, and then transformed into a syncopated rhythm, with the arrows showing the point of emphasis on the upbeat, on “and” before beat 3: