ESP01 unsuccessful code execution
If you are using an ESP01 I don't think you should be using GPIO1 as the output pin, its my understanding that they don't have GPIO1.

I think they have GPIO0 and GPIO2, so you should have attached the LED to GPIO2 and use digitalWrite(2, HIGH); in you code.

Have a look at this page and see if what its on about is what you are trying to do.

http://iot-playground.com/blog/2-uncategorised/38-esp8266-and-arduino-ide-blink-example