What are the implications for Boeing of its Dreamliner market structure?
The market structure for the Dreamliner is an oligopoly. This is because there are only a very few (probably only one other besides Boeing) competitors in this marketplace. This falls under the heading of oligopoly -- more than one seller, but not very many (a market condition with few sellers).
The implication for Boeing is that Boeing must be very aware of all of Airbus's actions (like price changes and changes in the aircraft or its marketing) with regard to the A380. Anything that Airbus does will affect the market for Boeing's Dreamliner. This means that Boeing has to be aware of what Airbus does and has to adjust its own actions in order to compete.
This is unlike how things would be if this were a market in monopolistic competition. In that case, Airbus's actions would not affect Boeing's market share in any significant way and Boeing would not need to monitor and respond to what Airbus did.