In a lot of programs around your Mac, if you need to insert a special character (like, say, the ellipsis or the trademark symbol), you can select Edit > Special Characters (Option-Command-T, where available) to bring up the Character Viewer.
Double-clicking on an item from that list will insert the selected character into your document, and away you go. However, if you use a certain symbol all of the time, it's worthwhile to know the keyboard shortcut for it. Luckily for us, our Macs have a built-in way to figure out what those shortcuts are, so here's how you find them. First, you'll need to enable a special menu bar icon by going to System Preferences > Keyboard, clicking on the Keyboard tab, and then toggling on Show Keyboard & Character Viewers in menu bar.
After you do that, you'll have this fancy new icon in the upper-right of your screen:
Click on the icon, and you'll see two main choices—not surprisingly, they're to open either the Character Viewer or the Keyboard Viewer.
The Character Viewer is the same window that we invoked before by choosing Edit > Special Characters, but the Keyboard Viewer is pretty cool. At first glance, it looks just like whatever your keyboard layout is.
But if you start holding down modifier keys, you'll see the symbols change. For example, here's what happens when I hold down Option:
What this means is that, if I hold down Option and then press the key specified in the layout above, I'll get the symbol pictured. So Option-; is the ellipsis, for example, and Option-2 is the trademark symbol. Nifty!
Here's what's revealed when I hold down Shift and Option together:
Shift-Option-K is the Apple symbol! That gave me a little fangirl thrill. Anyhow, this could be very useful, as a lot of the most commonly used characters are available as shortcuts.
Additionally, you may have noticed that my second keyboard screenshot above has some orange characters in it. With those shortcuts, you'll be given the choice of typing just that symbol or that symbol as a diacritical mark above a letter. So if I hit Option-n within a document, this happens:
I can then either press the spacebar to insert just the mark by itself, or I can press an appropriate letter to attach the diacritical mark to.
Of course, if you're running Lion or Mountain Lion, you can press and hold on a particular letter to see and select any accents available for it, as I've mentioned before.
I use the Keyboard Viewer all of the time, though, which is sad since the ellipsis is generally the only special character I ever use. So essentially I'm opening the darned thing just to remember Option-; at least five days a week. This is why they call me Ol' Swiss Cheese for Brains. At least, that's what I think they call me. I can't really remember.