In hair terms, an extension is a length of fake hair (or real hair for the very wealthy among us) attached to your own hair, to, well, extend the lenghth of your hair. Hair extensions can also be used to create false dreadlocks.
Extensions can be attached to hair in several different ways: by melting a small part of the fake hair and thus welding it to the hair, by weaving it into the hair... There is another way that I can't remember right now.