Darboux and Riemann Sums

Question

Can someone explain to me the concepts of Darboux and Riemann Sums? I find it hard to understand the true meaning behind the mathematical notation. For example, how can someone grasp this concept $$f\colon [a, b] \to \mathbb{R} \text{ is Riemann integrable iff }$$$$ (\forall \epsilon)( \epsilon > 0((\exists \delta) (\delta > 0((\forall \Delta\text{ partition of the interval }[a, b]) \implies (\exists I \in \mathbb{R})(\lvert\lvert \Delta \rvert\rvert) \implies (\lvert \sum_{i=1}^n(f(\xi_i)(x_{i}-x_{i-1})) - I\rvert < \epsilon)))$$