Emissions are basically chemicals in exhaust gases that are harmful to air quality, mainly carbon monoxide (CO), hydrocarbons (HC), and nitrogen oxides (NO). Healthy engines produce fewer of emissions, and older or "unhealthy" engines produce more. Some engine designs--like rotary engines, for example--produce more of certain emissions than others.
The oxygen sensor is a measuring probe for determining the amount of oxygen in the exhaust; actually they are mounted inside of your exhaust pipe. Oxygen sensors does their job by continually monitoring exhaust gases and signals the engine computer to adjust the air/fuel ratio to make sure that gasoline combustion is as complete as possible, thus reducing harmful emissions while also improving engine performance and fuel efficiency.
This is like an air filter in your exhaust system, only this "filtering" is done chemically by taking advantage of the extremely high exhaust temperatures. These can go "bad" if your air-fuel mixture gets way out of whack, acting like a cork in the exhaust. Like so many other automotive parts, catalytic converters are really only noticeable when they stop working properly. Catalytic Converters are expensive to replace. OUCH!
Another initialism (an automotive industry favorite), the EGR stands for Exhaust Gas Recirculation, an emission control device that helps make the ocean and skies blue and the grass green by re-circulating some of the exhaust gases back into the engine for another burning.