I've heard commentators call certain songs and artists "urgent" or say the phrase "he plays with such urgency" I have no idea what this describes.
"To play with urgency" is related to tempo, or time.
It describes music being a little prime on each beat, producing a feeling of urgency, of a fast driving movement. It could be a tempo indication by the composer, or a band being a bit stressed out, or an artist always producing fast-paced songs.