How does integration work?

There are a few different types of integration, the most common being Riemann integration.
Riemann integration allows you to find the area under the graph of a function between two points. Its definition uses something called a limit and it basically says we can approximate the area underneath the graph by adding up areas of rectangles (which is trivially "base times height") so that the width of all the rectangles together go from the first point to the second point and the varying heights of the rectangles goes from the bottom up and "hugs" the curve as best as possible.
Imagine you have a curve and approximate the area underneath it between a and b by finding the area of two rectangles each of width 
(b-a)/2, we realise it's quite a poor approximation, but if you make the approximation with 10 rectangles each of width (b-a)/10 and lots of varying heights we realise the approximation is better. Newton and Leibnitz (the independent founders of calculus) then said "what if we take the width of the rectangles to be really really small, so small that the width of the rectangles approaches zero!" then we realise we'd have an infinite number of rectangles to add up all of varying heights and all of width "essentially zero". 
What is really going on is that we've said let the width be dx and let us add up all the areas of rectangles as the limit of dx approaches zero. This can be seen in the standard lay out of integrals:
f(x) dx is simply the height of the function multiplied with a width (giving rise to the area of a rectangle) the integral sign at the beginning is an elongated S for "sum", hence you sum up all the areas of rectangles.