The difference between an embedded system and a general purpose computer system is one of purpose, and to a much lesser extent, design. While a general purpose system can be used for many things, an embedded system is only meant for one purpose.
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General Purpose Systems
A general purpose computer system is what you think of when someone says the word "computer." The defining feature of a general purpose computer is that it can be reconfigured for a new purpose. In the early days of digital computers, this involved actually rewiring the entire system. Today, most end users aren't even aware that this is happening, as the process has become completely transparent.
An embedded system is a little harder to pin down. It is dedicated to a single purpose, or a small set of purposes. Embedded systems can be found in nearly every single piece of modern electronics--in fact, they are the electronics. A modern television, a portable music player, a computer-controlled air conditioning system or virtually anything made in the last 10 years that isn't a general purpose system and requires electricity: that is what an embedded system is.
Some devices don't fit firmly into either camp. Smart phones are the best example of this. While they use a design philosophy that mimics an embedded system, they can also function as general purpose machines. They may have a single overarching purpose (they are cellular phones after all), but they are not limited to that purpose.