Hi Jay,
Welcome to JavaRanch!
Your problem is that you've got two different copies of the variables "exitButton" and "picture." One set are member variables; the only place these are used is in the actionPerformed() method. The other set are local to the PhotoAlbumApp constructor; these are the ones that point to the actual buttons. So when the ActionListener asks if the source of the event is the member variable exitButton, which is null, the answer is no, so the code doesn't run.
To fix this, you'll want to change the two lines
to look like
Then you're not declaring new local variables that hide the members anymore; you're just using the members.
If this doesn't make sense, then you need to back up a little and learn more about the Java language before you try to write this program; check out
Sun's Java Tutorial, especially the first four sections.