Many different physical phenomena are modeled with partial differential equations. One of the most common is how heat propagates through a substance over time. This is a very typical example you will encounter in a class covering partial differential equations.

Fundamentally Engineers and scientists of all types need to understand this math because many physical systems in chemistry, mechanics, electronics, and more can be modeled with and thus understood with partial differential equations. These equations allow us to ask questions about and understand these physical systems. And perhaps even more importantly let us make predictions in the future. For example, going back to our heat transfer problem, how big a heat sink does your computer processor need to keep it from melting? Some engineer solved this problem for your computer using these equations or the results of them.

Sure our computers do a lot of this math automatically so you don't have to think about it too much day to day, but anyone who is going to be a real expert in just about any phsyical science or engineering discipline should have at least a basic understanding.

Wikipedia has a pretty good page on the topic which I've linked below.
Source(s):
http://en.wikipedia.org/wiki/Partial_differential_equations

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