Two power source for Arduino and a relay In my work, I have an ESP-8266, Arduino-Uno and a Relay. ESP 8266 and Uno is connected. Relay's data signal is take from arduino's 12th pin. ESP is powered from Uno.

If I use two different power sources for Uno and Relay, Relay does not response for the signal from Uno. (Both have a same common ground)

If relay is powered from Uno, then after load control (power on and off) ESP-8266 goes down.
The uno can't provide enough power to power an ESP8266 reliably.  I suggest you power it from a separate source.

When ever you use separate power supplies for a circuit you need to "tie the grounds together", connect them.  Otherwise the circuits don't use the same definition of zero.