The keyboard offers a number of common characters for the language layout you have chosen, especially when you use modifier keys such as Shift, Command, and Option to invoke alternative character sets; however, while those that are mapped to various keys are common, they are only a minor subset of the characters that are available for you to use when composing documents.
OS X's support for unicode fonts makes a vast array of characters available for you to use in applications that are coded with Apple's Cocoa frameworks, and while some programs have their own input methods for accessing these characters, the system has a browser through which you can find and use almost any of them, including math symbols, Greek and Latin symbols, Chinese characters, emoticons, and even Braille.
These symbols are available in the system's Character Viewer palette, which is disabled by default in the system, but which can be enabled by going to the Language and Text system preferences in Lion (or International in other versions of OS X), selecting the Input Sources menu, and then checking the boxes next to the Keyboard Viewer and Character Viewer. After doing this, check the box for the option to show the input menu in the menu bar, and you will see a small input menu show up next to the date and time in the system menu.
The input menu is a black-and-white icon, but if you select more than one country input layout to include in the list, then the input menu will change to show the flag of the currently selected layout, a feature that some people may prefer to just the basic input icon.
With the menu enabled, choose the Character Viewer option in it, and you will see the character viewer window appear which has columns showing categories, items in the category, and then related characters. By default Apple only shows a few of the common categories such as emoticons and Greek symbols, but you can enable others by clicking the gear menu at the top of the window and choosing "Customize List..." In the list you can check various additional categories to include in the character viewer window.
When you find a symbol you like, you have several options for managing it. To insert it into your document you can either double-click it, or you can drag it from the palette window to the desired location in your document. If the character is one that you will use again, then you can click the Add to Favorites button and it will be placed in the favorites section of the character viewer for easy access. In addition to the favorites, a group of the most recently accessed symbols will be kept in the Recently Used section so you can find them there. This feature is useful especially since many symbols are very similar and finding the exact one you used in a previous instance can be a bit difficult at times.
A final feature of the character viewer is its search function, which is a bit unique. If you search for a name such as "heart," then all the characters will be input independently so you can find similar characters to all of those that make up the word "heart." In addition, the character viewer will show various symbols that either include or represent hearts. Unfortunately the search is not always complete, so if you want a spider dingbat character, you can't find it simply by typing "spider," but in many cases the search is close enough.