Mail app won't close. How do I force it to close?

Control-click on the Mail icon in the Dock while pressing the option key. Choose "Force Quit".

If Mail is still synchronizing with a remote mail server, or anything funky happens to the connection between the remote mail server and the Mail client, then there is a short period of time that Mail waits for the network connection to time out. Either of these conditions will supercede a shutdown request, and you can either wait, or use the last resort suggested by dialabrain.