Lumia 820 comes with a 1650 mAh battery as standard and it is marked BP-5T.
If you want more juice, do a web search for "Lumia 820 high capacity battery" and check prices, reviews and details.
I like Mugen Power. I've used them before. They have a slight boost at 1800 mAH and that does not require a larger back cover... Anything like 2500 or 2700 will require a larger cover or a "battery case." You won't get that much extra juice without a big increase in battery thickness. If you are going big - go all the way. Mugen has a whopping 3600 mAh battery with an enlarged cover in black, white, yellow or red.