its your main relay. it controls the fuel...when its not working properly your car will turn over because your starter is turning the motor but it wont supply any fuel. its a very commong problem w/ accords especially w/ 4th and 5th gens but now i guess the 6th gens are affected too.

probably wanna replace it or take it out and find a good electronics repair shop to clean the circuit board and re solder all of the connections.

its located on your steering column under the drivers side dash.