Color identity is a concept expanding the color of cards. While in game play mechanics the color of a card is always determined exclusively by the colors used in its casting cost or its color indicator (which was retroactively applied to cards like Ancestral Vision, which previously specified it in the rule text), the color identity of a card is specified by all mana symbols that appear on the card anywhere, including within its rules text. This is primarily relevant for the Commander format, in which players are not allowed to use any cards whose color identity isn't encompassed by that of their General.
For example, while Fires of Undeath is a red card, and Silver Knight has protection from it and White Knight does not, it can't be used by a player who uses Kumano, Master Yamabushi as his or her General in a game of Commander. This is because the color identity of Fires of Undeath is red-black and is not a subset of the color-identity of Kumano (red).
Phyrexian mana is considered to be the mana of the color it replaces, e.g. Dismember is a black card. Hybrid mana is considered to be either color, thus Kitchen Finks has a color identity of white-green.