I pray to kindly show the intuition behind the concept of rough path. Google provided some links that deal with the notion of rough path but was difficult for me to have an idea.
A rough path is an interesting analytical and algebraic object:
algebraic: it's a multiplicative functional in a (truncated) tensor algebra obeying Chen's Theorem (non-commutative analogue of the additivity of integration - Chasle's relation).
analytic: it's of finite p-variation
the best place to start is to read T. Lyons et al's St. Flour notes "Differential equations driven by rough paths".