PieterP:
Who said that?Of course you can use it "on its own", power-wise, there shouldn't be a difference between using it stand-alone or as an AT slave connected to an Arduino.
It's normal for the module to get warm when using the WiFi radio, but it shouldn't get too hot to touch.
This Beginner's Guide to the ESP8266 should contain everything you need to know.
Pieter
Ah right. Thanks.
pert:
No LED is shown at that link. Try again.
I couldn't think of a better word for this. It's not a LED, is it?