Is there a website that I can get a cheap plane ticket from Frankfurt to Dusseldorf?

Due to the distance (about 230 km) between Frankfurt/Main and D眉sseldorf, no airline offers any service.
The train (www.bahn.de) is your next best choice, regular service (about 1.5-3 hrs) at a cost, depending upon day and time, is anywhere between 46 Euro-65 Euro. Weekend specials are 30 Euro, but you have to take local trains only and it takes about 4hrs.
The lowest priced choice is http://www.mitfahrzentrale.de/, it's a share-a-ride service and cost is approx 11 Euro and lower, depending upon how many people are sharing a ride in the same car.

http://www.ryanair.com

Frankfurt, Dusseldorf, are you from earth?

Fly? The cities are so close.
If you take the 8:38 train, you arrive in Duesseldorf at 10:14, about 1.5 hours. Just getting to and from the airports, let alone check in and security would take you much longer.
You can probably get a cheap tickets for around EUR40, even lower if you are flexible with your time.
try http://www.bahn.de/

Well by train you make 3 hours. But if you insist take a look here...

www.condor.de
www.airberlin.de
www.ryanair.de
www.lufthansa.de

And if you are stil curious:
www.dbahn.de

Hey, give you a tip, never done it, but heard of many friends that do it. There is a "ride service". You register and someone that is going to drive this way takes you with him/her. You pay a part of the gas and you get to meet new people. If i get the infos i`ll write back. Or just ask here, someone should know about this stuff.