A Riemann Sum for ∫ab f(x) dx is when you chop up the interval [a,b] into n subintervals [ai,ai+1] where a = a0 < a1 < ... < an = b and pick an xi in each interval. You get rectangles of width ai+1 - ai and height f(xi). The sum of their areas approximates the integral.
To get the exact value of the integral from the Riemann Sum, you take the limit as follows:
1. n → ∞ (Large number of subintervals).
2. maxi=1,...n(ai+1 - ai) → 0 (All subintervals have small length).
This method of computing the integral is called Riemann Integration, and is used as the definition for Integration in Calculus 1 and 2, since it is sufficient for integrals of continuous functions to be well defined.