I saw that my Natural Gas provider stated I could receive a benefit if I had one or installed one, so I was wondering what it was.
In this context a boiler is more-or-less the same thing as a tankless/on-demand/instant water heater. From what I can gather the term 'boiler' is used instead of one of those other terms based on characteristics of the unit or the usage. Heaters with higher BTU (or kW) ratings, or designed for higher duty cycles, or used as a heat source in a space heating or space heating plus domestic water heating application are often called boilers.
AFUE is annual fuel utilization efficiency. It is a measure of heat output (in the heated water) vs heat input (heat capacity of the fuel gas being burned). Higher AFUE is more efficient.