Using Arduino Uno R3 + Ethernet Module, how to control current pass to Lamps

An Arduino Uno can be powered by

• a stable (regulated) 5V DC, which you can either supply via the USB power lines, or via the shield connectors, or
• an unregulated 6-20 V DC (7-12V recommended), which you can supply via the 2.1 mm centre-positive barrel plug connector.

I would not recommend using a 9V battery, because its voltage will drop quickly, but in a pinch this can be used.

ref: http://arduino.cc/en/Main/ArduinoBoardUno

Obvious problems: Well yes. As drawn right now, there is some confusion but:

1: Either you don't get 220V to your secondary relay (The one connected to the Arduino) because the line leaving your primary relay going left is a trigger signal line, rather than a power line OR

2: You DO get 220V to your secondary relay and you cannot trigger your first relay because the line leaving your primary relay going left is a power line, rather than a trigger signal line.

My first recommendation would be to redraw your diagram with clear labels on all wires.