I have the following set up between a nano and the ESP-8266:
VCC-------->+3.3v
If that's ESP8266 VCC to Nano 3.3V, be aware that the nano 3.3V regulator is not powerful enough to power an ESP8266.
If that's Nano VCC (5V!) to ESP8266 3.3V, that's just wrong and has probably fried your ESP.