A connection scheme I hope everyone can understand:
Use a 5v power supply with an adaptor , use 2 jumper wires to connect Vcc to Vin in the esp8266 and GRD to G there.
The relay in the picture is 3v so connect the 3.3v supply to the relay's Vcc and the ground to GRN. Connect any digital pin that will control the relay, in the picture I used D7.
The wiring to the AC device, in the picture a bulb : it has two wires (not counting the earth, usualy green or yellow), leave one wire connected and cut the other and make an extension through the relay AC side. Use the NO pin and the COM to connect to the split wire of the lamp. The NO (normally open) pin will connect with the com when the control pin D7 is true.
Before you connect to AC power - isolate the connections to COM and NO so you will not touch it by mistake !!!
Connect to AC power for a short time when control is on and take the AC plug out, then check the relay's temperature. If it is hot - the power that goes through the relay is too large and you need a higher level relay (larger current).
If you use a 5v relay you need a logic converter to make the control signal from D7 a 5v instead of 3.3v.