EDS/EDX stands for Energy Dispersive X-ray Spectroscopy. It's a very useful analytical method to determine the elemental composition of the a very small area of the material of interest. An EDS mapping helps to determine the elemental distribution of the area of interest.

The EDS analysis can be coupled with an imaging tool which used collimated beam of electrons for imaging. Such imaging tools are SEM, TEM or STEM. The collimated beam of electron if impacted with a material, it excites an electron in the inner shell of an element to escape from its initial location creating a electron hole. Another electron from the outer high energy shell jumps to the low energy shell to fill in that gap. The difference in energy in between two discrete energy levels is then emitted as X-rays, which are characteristics of an element present in the sample. The energy of the x-ray can be measured using an energy-dispersive spectrometer. As the atomic structure of an element is unique and the difference in energy in between two shells are known, an elemental composition is obtained by EDS.

More information about FIB is available in Wikipedia. http://en.wikipedia.org/wiki/Energy-dispersive_X-ray_spectroscopy