That's excellent, Melinda! :) Just a small addendum - think of integration as breaking something up into a very large number of very small bits and adding the results together. For instance, in the case Melinda cited, a variable force - let's say you're trying to figure out how much work gravity does on an object falling from a certain height. Well, that's easy - work is force times distance, so you just multiply the weight of the object by the distance it falls, and boom. No pun intended.
However, suppose the height is so great that gravity can't be treated as a constant - what do you do then? What you might do is break it up - say, figure out what happens for the first quarter of the fall assuming gravity is constant, then refigure gravity and do it for the next quarter, and so on until you get it to the ground. Then you add up the four pieces.
That'll certainly be better than if you took it all at once, but it's still only an approximation. How do you know if the approximation is good enough? Clearly (and this can and should be proved), the smaller the pieces you take, the better your approximation will be. It seems reasonable to suppose that, if you could take an infinite number of pieces and add them all together, there would be no error. This is what integration does, in concept anyway - break the distance into infinitesimal pieces and add the results all together. Any time you have a situation where you're trying to figure the effects of something which varies over a range with which you're concerned, be it distance or time or anything else, integration is the primary tool for analysis.