What's the dot in the centre of the red window close button for?

Solution 1:

Specifically, it means that hitting that button will produce a prompt rather than immediately performing the action of closing the window. While this is usually a save dialog, it could be anything requiring further user action.

In this sense, it is equivalent to ellipses on the ends of menu entries.

Solution 2:

In most apps, it's an indicator that there's an open file with unsaved changes (although this is less common now that many apps use OS X auto save). In Terminal it indicates a process is running in the window.

Solution 3:

I like to think that it's a gentle remainder: "Something is going on here, and if you close this window, whatever is going on here will be interrupted or lost".

On most apps, it means that the file has been updated (it's "dirty"), and that you will receive a prompt asking if you want to save it or not. Talking specifically about the Terminal, it's telling you that there's a task running, and closing the window will kill the task.