A scoring situation that can only be accomplished after a touchdown has already been scored. A two point conversion results when an offensive player crosses the goal line into the endzone, either by running in or catching a pass in the endzone.
Generally teams opt to kick an extra point instead of going for a two point conversion because two point conversions have a much lower chance of success statistically. Two point conversions operate much like any other regular redzone (inside the 20 yard line) play. All running and/or pass plays are permissable. If the two point conversion fails the team still gets six points from the touchdown they just scored, but now they only have six points instead of the seven they could have had if they had kicked the extra point. Most teams will use the two point conversion if the game is close and they need just a few more points to win.