I think it might be easier to present the failure diagrams first. There are two main failure diagrams.
With the above failure diagram, it is now easy to explain this failure diagram. here is greedy, and is the correct way to kill. It transposes back to the previous failure diagram.
Note that must not be played at a, which I leave as an easy reading exercise.
and are miai for life, so Black lives.
The main line is this diagram. connects and allow to live, and both parties are happy. This should be what is played in real games.