Thanks for all support,
When I am using USB's alimentation my relay is functionning well. But, when the alimentaion is coming from the arduino my RELAY ISN'T WORKING WELL.
What the heck is alimentation??? You mean power? Or voltage?
Do you have the Grounds connected?
The schematic is a big mess so pretty useless. But it does show:
- You should use a base resistor
- You should use a fly back diode across the relay coil
- DON'T connect mains to GND of the circuit. Keep it nice/safe and separated.
alimentation means "feed" in German, I think. 
The schematic is fine, very easy to follow.
The first regulator seems to be a 7815, powering the Op amp.
That means your op amp high out will go to 15V. A0 input clamp diode will do its best to limit that to 5.5V, so your readings may be off there.
NPN base resistors needed, yes. 270 ohm to 1K should do.
Diodes across the relay coils, yes. Anode to +5, cathode to collector of transistor.
Be sure to connect the Arduino GND to the other GNDs.
The AC going thru the relay contacts, DON'T connect that to the other GNDs. That should likely go to the AC neutral of your AC supply, or however you are wired up overseas for your AC powered appliances. Treat the relay like a switch in the return line.