Esp 8266 doesn't reply to AT commands
Try powering the ESP from an external 3.3V power supply if possible - some people say the Arduino 3.3V regulator may not provide enough power for it (it is quite power hungry at startup).  
Another option worth trying is to put a 100uF capacitor between ESP VCC and GND. Put some small delay at the beginning of your setup() function if you do that just in case.  
Also, there are several pins you need to pull UP: CH_PD, RESET, GPIO0. If you have GPIO15 broken out (depends on the board), pull it to ground. I have also noticed the ESP won't boot properly if GPIO2 is LOW - either leave it floating or pull it up.  
Here is a picture you can use as a guide  
Of course, you should make sure the ESP is flashed with the AT commands firmware (they usually are) and your baud rate is correct. If you don't know - try 9600, 57600, 115200.  
If you have a USB to TTL converter, it might be better (easier) to test your setup using it, without Arduino. Any serial terminal program (including Arduino IDE Serial Monitor) will be fine. Just make sure you send both CR and LF for new lines. If you send "AT" and receive "OK" your wiring is correct, if you don't - it is not.