I use Mac OS X Terminal.app. But the answer to this would probably apply to any typical UNIX-ish terminal emulator in a graphical environment.

Question -- what do I do to: Use my mouse cursor to click on a character-position in the current line of the terminal, and have the terminal's cursor jump to that spot?

Typically, you have to hold down an arrow key to move to the correct cursor position. If you're pasting in a long string of text at the shell prompt, or working in an editor like VI, this can take a long time. I know editors have other navigation keys like jump-words, but I like my mouse cursor.