"Sie isst nichts."
"Nichts" stands where we might say "not anything, nothing", whereas the sentence "Sie isst nicht." would imply that she does not eat (like, ever...it's not something she does).
In another example, say I am at a party and not thirsty. Someone may ask what I want and I'd say "Ich trinke nichts, danke" (Nothing for me, thanks). However, if I don't drink alcohol and someone offers me a beer, I might say, "Ich trinke nicht." (I do not drink).