Arduino Nano v3.0 with ESP8266
The Esp8266 module has different firmware, the old one was running at 9600 and the new one is running at 115200.

Try to upload this code on  your arduino Nano and send AT commands over the Serial monitor of the Arduino.

If you send AT it should returns "OK", if not, try to swap RX to TX and try to play around with the baudrate 9600, 57600, 115200.

It might be that your Wifi module doesnt have any firmware loaded which might need to be done.

#include &lt;SoftwareSerial.h&gt;

SoftwareSerial wifi(10,11); // RX, TX

void setup()
{
Serial.begin(9600); // FTDI
wifi.begin(9600); // GPS; 9600 first time, 38400 second time
}

void loop()
{
if (Serial.available())
wifi.write(Serial.read());
if (wifi.available())
Serial.write(wifi.read());


}