AltGr is short for "Alternative Graphics". On many keyboards, you will find the key to the right of the spacebar labelled "AltGr". In many cases, even if this key is labelled Alt, it will function as an AltGr.
This key is used in a fashion similar to the shift key. You hold down AltGr while typing another key. e.g. You may hold down shift and press '7' to create an ampersand, at least on a U.S. keyboard. Similarly, you may press AltGr-E (or -4, or -5 depending on locale) to create the symbol for the euro. These alternative symbols are normally inscribed on the keys of keyboards containing an AltGr, just as with the ampersand above a 7 on a U.S. keyboard.
By default, both Alt keys on U.S. keyboards act as Alt. However, if one changes his input locales under the keyboard control panel, a number of the European configurations use the right Alt as AltGr. These layouts can make it easier to enter some graphical characters such as © and €. Similarly, a resident of Europe can configure her input locale to make his AltGr key to act like a simple Alt key. The drawback in both cases is that the characters printed on the keyboard no longer match those typed.
Graphical or extended characters can be entered in a more difficult manner in Windows (and DOS for that matter). One can press and hold the Alt key, press a series of numbers on the numerical keypad, and finally release the Alt key. See more under compose key.