The AltGr key functionally resembles the Option key. This key is placed on the right of the space bar. In European keyboards, the AltGr key has the text “AltGr” or “Alt Gr” in the keycap. In US keyboards, it looks the same as the normal Alt key and is often called “the right Alt key” as opposite to “the left Alt key”. This reflects its role in the normal US keyboard layout: it is just another Alt key, for use when you find it more convenient to press Alt with your right hand (similarly to the duplication of the Ctrl key and the shift key). But in many other keyboard layouts, including most European layouts, it can be used to produce various accented letters and other special character—whence its name, it is short for “alternate graphic”.
The Microsoft page Windows Keyboard Layouts contains information about a large number of layouts. It shows them as small virtual keyboards, and you can see which layouts use AltGr just by looking at the key to the right of the space bar: is it labelled Alt or AltGr? If it is AltGr, clicking on it shows the characters it can be used to produce.
You can enable different keyboard layouts and define shortcuts (e.g. Ctrl 1, Ctrl 2 etc.) for switching between them. That way you can utilize different AltGr combinations.
Even a common US keyboard can be made to use AltGr, e.g. by using the US International layout shipped with Windows (which is rather artificial, but still useful). If you need mathematical symbols, check out my Math keyboard layout.