An embedded system is a particular device that combines hardware and software to perform a particular task. A great example of this is a camera; it is hardware combined with software that stores pictures and has additional features to enhance images.
An embedded system engineer is responsible for developing and implementing software for products with embedded systems. They can also work in the hardware-side of the development process, but their responsibilities will vary depending on the organization. An embedded system engineer is also required to maintain and update software.