As far as I know - 'Noise' is somewhat equivalent to grain in traditional 35mm - which is what you get (extra grain) when you use high ISO traditional film - so traditionally a grainy picture is one where you see the grain structure of the film. You can see the grain better in an enlarged 8x 10 inch print than you can in a 4x6 inch print.
Noise is very similar in digital photography but instead of grain (round) you see pixels which are square. You usually get noise when you set the camera's ISO too high or you are shooting in low light without flash. There are other components and facets to noise...but it gets even more technical.