What is the difference between emotions, intuition and affect? From what I know, intuition is affect-based. But how do they differ?
Affect is a fundamental feature of consciousness and is a mental representation of input from the body to the brain.  Affect corresponds to feelings of (un)pleasantness and activation (arousal). 

Emotions are conceptualizations (interpretations) of your affect in a given context.  You might feel pleasant and highly aroused (an affective feeling) on a roller coaster, and your conceptualization of that affect in that context would be "excitement" (an emotional feeling).  In this sense, emotions are what we would traditionally consider to be "cognitive."

Source

Under this scheme, intuition doesn't really have a direct, formal relationship to affect or emotion, although intuitions are often affective (well, pretty much everything is affective).  But see here for a more informed discussion of the literature.