ESP8266 Arduino adapter
The ESP8266 WiFi to UART module works with 3.3V, so to use it with Arduino UNO (working with 5V), it needs a simple adapter. First it needs a 3.3V power supply. It is recommended to use an external 3.3V voltage regulator, because the 3.3V pin from Arduino can not give too much current. Then the WiFi module also needs a level shifter on the RX pin, which can be made using two resistors. The TX pin from ESP8266 does not need a level shifter, because Arduino UNO "understands" 3.3V as HIGH level for digital input. A series resistor of 220 Ohm on the TX pin can be used for safety, for the case when the Arduino RX pin somehow gets to be an output.
Note: for normal operation of ESP8266, GPIO0 and GPIO2 must be floating or HIGH at power on.