If your wondering what the AIR and EGR systems do, read the following. Thank Matt87GTA from MNFbody for the details.

The AIR system (pump) is, like Freak said, there to pump fresh air into the exhaust manifolds to lower harmful emissions. It does this in several ways. Basically the air being pumped into the exhaust manifolds "reburns" excess fuel on cold startups to lower the levels of Hydrocarbons (HC) in the exhaust and warm up the catalytic converters so they can reach their operating temps faster. Much like a choke on a carb'd vehicle, the cold start routines on EFI engines will inject more fuel into the engine during cold starts to aid in starting, staying running, and to warm up the engine and its emissions equipment (catalytic converters) faster. The extra fuel will be too much for the engine to fully burn in the cylinders and this results in unspent fuel being present in the exhaust. The additional air being pumped in by the AIR will react with that heated but unburned fuel and burn in the manifolds which warms the exhaust temperatures up significantly which in turn warms up the catalytic converters. Catalytic converters need to be warmed up to be effective at reducing HC levels and NOx levels and such in the exhaust. Once the catalytic converters are up to temp, the AIR will turn off (this is a preset routine in some applications but some (newer systems) will moniter O2 sensor activity and stop the AIR system once the catalytic converters show a reduction in emissions indicating they are up to temp).

EGR stands for Exhaust Gas Recirculation. This valve opens to allow small amounts of exhaust gas to flow into the intake manifold. Exhaust gas is inert and therefore cannot be a party to the combustion process. This effectively cools the cylinder since less combustion can take place on a given power stroke. The cooling of the cylinder serves to lower the chances of NOx being generated by the engine which are generally believed to be harmful emissions and can only be a byproduct of combustion at extremely high temperatures. The EGR also, as some of you might have guessed already, allows you to run more timing advance and require less fuel during engine operation without taking the chance of superheating the pistons and other parts of the combustion chamber. That means it will increase mileage and lower emissions - two things the OEMs are very excited about. The EGR will only operate, as nikos95 stated, at part throttle though and its overall impact on mileage is really quite small in the grand scheme of things.

For us (performance enthusiasts), the EGR is mostly useless. The increase in mileage is going to be measured in the tenths of a MPG usually and we really don't care much about emissions (even if we did care about emissions, the amount we produce running around at WOT all the time far offsets anything an EGR valve can ever do when we are at part throttle). And the EGR valve and all of its plumbing and wiring is annoying and ugly. A lot of newer engines do not even bother with an EGR system as the processing power of the ECMs/PCMs and accuracy of their engine management operating systems and calibrations is far beyond those of the engines in the past that really needed an EGR to be sort of efficient and not pump lots of NOx into the atmosphere. The AIR system is probably here to stay until they come up with a type of catalytic converter that doesn't require being heated up to do its thing.