AFIK
The ESP8266 has 2 low power modes, but I have to tried either.
I'm not sure if the AT command set supports either, but as the AT firmware is open source, its quite possible to enhance and recompile and reflash the device.
In fact, its almost impossible to use the device in the first place without reflashing it, because there is no way to know what firmware it comes with from China.
Of course if you buy a board from PaulWare he would put the correct firmware in th esp8266 I presume 
Anyway, back to power modes.
Deep sleep seems to be the most used method, and the API function causes deep sleep for a predefined length of time.
Theoretically you can also do wake on interrupt, but I've not seen any code that demonstrates how to do that.
From what I recall, on wake from deep sleep is virtually the same as power cycling the device, I.e I don't think internal ram vars are retained, but it is possible to store stuff in flash albeit with a limited number of write cycles.
I think there may be another less drastic low power mode, but I have not investigated it, so is could be wrong about its existence. Note there is a lot of conflicting information out there about the ESP8266
So take everything you read with a pinch of salt.