The Character Viewer gives access to every glyph available on your system. The Character Viewer can invoked from the same menu as the input methods (“keyboards”), but you need to tell the computer once that you want the Character Viewer there. To do so, open the System Preferences > Keyboard. Check the box to show the Keyboard and Character Viewers in the menu bar.
Now that you have the Character Viewer easily accessible, you can call it up to enter characters for which you do not know the key combination or that you may not be able to enter with your usual input methods (even if you do not exactly know what it is called). For example, if I wanted to type some unknown character with a caron on top, I can open the character viewer and search for “caron.” This will show me all characters whose official name has “caron” in it. (If you do not see a full window as in the image below, click the little icon at the upper right).
All I have to do to put this in my document is double-click the character in the main pane of the Character Viewer (assuming, of course, that my document was open with my cursor positioned where I want the character to go when I launched the Character Viewer). You can also see details about the character in the right-hand sidebar, such as that the name of the thing I want is called an ezh.
If you think you will need the character again, you can add it to a list of Favorites for easy access. If you need to refer to the character (e.g. for someone else to type it), you can also find the Unicode hex value here (see this post on how to enter characters using Unicode Hex Input).
You may find that the font you are using in your document does not include the character you need. Many applications will simply substitute a different font that does include the character, but this can make your text look mismatched. Fortunately, the Character Viewer also shows you what fonts on your system do include each character. This is under “Font Variation” in that right-hand sidebar. You can choose a particular font for your “special” character so that it will blend in better with the rest of your text; or you may decide to change the text of your document entirely to use a font that does include your “special” character.
Unfortunately, the versions of the Character Viewer in Yosemite and El Capitan force you to click through each variation to see the name of the font (the font name used to display under each variation of the character).
If you cannot find your character by searching, you can also just browse around in the Character Viewer by Unicode ranges or other categories. This is particularly handy for finding things like symbols and emoji.