I have a d-link wireless router that I want to plug my arduino into with an ethernet cable and let mobile devices wirelessly connect and communicate with the arduino. Let me rephrase this. The areduino is a stationary unit tied directly to the wireless router. The devices that need to communicate to the arduino are mobile and will connect to the router, wirelessly, and the router will rout that information to the arduino. Is this possible? If so, please lead me in the right direction. Google isn't helping me out on this one 
It is possible. What model D-Link do you have? It should route by default. Find the subnet for the ethernet. Assign the ethernet shield a static ip in that subnet that is not assigned by the dhcp server, and plug it in. Then use that ip from the wireless.
Basically, you need an Ethernet shield for your Arduino.
I knew I'd need an ethernet shield but which one should I get? I have a mega 2560.
&& Model: WBR-1310
You need one that is within your budget and fits your Arduino and supports the Ethernet standard supported by the router you're planning to connect to. So of course you need to know what type of Ethernet that is. And then you need to know that there is a driver library for that shield that is compatible with your Arduino IDE version and your Arduino board type. And then you need to learn how to use that driver library, and then design, code and test the sketch that uses it to do whatever it is you're trying to do.
The below w5100 shield for $10 shipped might be a choise.