Russian word "нет" is interesting. As far as I know, it is a reduced "не есть" (is not), and it had the following evolution: "не есть" -> "несть" -> "нет".
However, the "не есть" itself is still present in the language, with "есть" skipped as usual. In the contemporary Russian, "нет" and "не есть" mean two different things:
- "нет" means "no" or "[there] is no". In the latter case, it requires an object in Genitive. Example: "there is no spoon" = "ложки нет"
- "не есть" means "is not". Example: "I am not an American" = "я не американец/американка"
The rule of thumb is: "нет" = "no", "не" = "not". Hope this helps.