EGR = Exhaust Gas Recirculator (may not be the exact wording, but close)

The EGR is for emissions purposes. At anything under WOT it will allow some exhaust back into the intake manifold in an efforts to reburn unburned fuel and help with emissions (at least I think that is its purpose). Under wide open throttle a properly functioning EGR is completely closed, and no exhaust is recirculated back into the intake manifold.

Most of the time eliminating it doesn't add power, just simplifies things under the hood (particularly if you are doing headers). In the case of the Focus, the poorly designed EGR opens too quickly under partial throttle causing lurching/hesitation. Most of the Focus guys eliminate this only to improve driveability, not performance. With the newer Ford's you do run the risk of throwing a Check Engine light when deleting the EGR.