Custom Arduino circuit not working
Your reset signal is tied low. The bar above it indicates that it is an active low signal. It needs to be pulled high to avoid resetting constantly. However, there should also be a button there so that your processor can have the reset signal pulled low (by the button) in order to reset it in the event of a lock or something. The reset button would have the same configuration as your switches do.

You can easily fix your problem by cutting the trace to ground and attaching a wire to 5V, so you don't have to make another PCB. You should update your schematic though so the next time you have your design fab'd you don't make the same mistake.