18650 Li-Ion, standard Li-Ion batteries - if you're only using it with a semi-permanent mains supply, you MUST ensure that it's charging requirements are met (you can't just leave it on trickle charge, as it's likely to explode and set on fire).
I would also suggest a 1 Farad supercap across the supply near the chip, as the high current pulses can occasionally dip the supply enough to reset the SIM800.
As I'm using battery power only I switch the power to the SIM800 using a dual FET switch, so as to minimise current when not transmitting. On a single 18650 I've recently run it until the battery was too low (for testing purposes), it lasted just over 4 months transmitting every 8 hours, a total of 378 uploads. I've recharged it, and put it back running again - and it's not even a new 18650, it was out of an old battery pack we replaced.
The actual ones 'out in the field' have two 18650's in parallel, and only transmit weekly, so apart from self-discharge they should last a LONG time - we're waiting to see what happens (among the required data they send, they also report the battery voltage).
For long battery life shut down everything that's not needed, and just run a 32KHz clock that wakes it up every second - mine wakes with signal input as well, but for checking mains there's no need - just use the 1 second clock trigger and check if the mains is present, and react accordingly.