BTW, I always thought of convolution as multiplication. Anyone else feel cheated by this guy for that reason?
You can often describe the exact same thing using a discrete sum or an integral, e.g., in a classical mechanics class you first consider a system of N connected springs, and a number of properties (like, say lagrangian, energy etc.) of the system will be expressed as sums with N terms. Then you take the continuous limit, the connected springs become one string and voila, all sums become integrals.
In quantum mechanics for some systems the set of all energy levels is discrete (e.g., the harmonic oscillator), for some it's continuous (e.g., a free particle), and often it is part-discrete part-continuous (a hydrogen atom.) Many formulas will require you to take a sum for the discrete part and an integral for the continuous part. Some quantum mechanics textbooks introduce a special symbol, a capital sigma overlaying an integral symbol, which means "integrate or add, whichever appropriate."
On convolution, it's an integral and can be thought of as a type of multiplication too, right? :)