That depends on which Call of Duty you are referring to. A zombie mode is present in Call of Duty Black Ops and COD World at War. You don't have to buy anything other than the game, these modes are included.
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you need to buy it
It is illegal to download "Nazi Zombies" for free. You can buy the disc for "Call of Duty World At War" and "Nazi Zombies" is on it.
No Zombies are a development of Treyarch for Call of Duty World at War and Call of Duty Black Ops and is not part of Call of Duty Modern Warfare 2 designed by Infinity Ward
Call of Duty : World at War and Call of Duty : Black Ops both have nazi zombies modes.
It's not available for the Modern warfare series of call of duty