Charging the stock battery for the drone 2.0 is done by the 4 pins built into the case. Otherwise, it is a regular 3 cell lithium polymer battery with a "mini tamiya" discharge connector. I think the drone 1.0 had a different charging mechanism.
You can buy batteries that are specifically designed to be compatible with the drone and the drone 2.0 charger (i.e. with pins in the case) - this is the easiest option. You can tell these because they will look very similar, and will only have 1 pair of wires and connector attached to the case. Bear in mind that if you go this route (i.e. use the stock charger), charging time will be much longer for a higher capacity battery. A 2000mah battery won't allow your drone to fly twice as long because it weighs more, but it might take twice as long to charge. It will also affect performance.
Personally, I would stick with one close to stock weight.
The other option is...
LiPo batteries are widely used in the RC world, so are cheap and very easy to find but not as convenient if you don't already have RC equipment. Almost all RC lipos have two connectors:
One for discharging at high current - this has two poles/wires, but the connectors used vary. XT60 is popular now and is what HobbyKing pushes, it is a distinctive yellow colour.
One for charging - a "balance connector" (usually JST-XH) that provides a connection to each cell independently. A 3 cell battery has 4 wires.
If you go this route you will need a lipo charger and probably a mini tamiya adapter.
Most modern RC lipo chargers are powered from a 12v-ish power source (either a car battery or a 12V AC/DC power supply like the one earlier in the thread). These typically charge by connecting both of the battery's connectors, which allows them to charge at much faster rates. You can still buy some very cheap AC chargers that only use the balance connector, but these usually charge very slowly. Be mindful that lipos are very dangerous, and charging them correctly is especially important.
Choosing an RC battery will basically be a factor of weight (including connectors of course) and size. It MUST BE 3 CELLS but I would not worry too much about the "C" discharge ratings - the drone doesn't draw a huge amount. Don't exceed the charge rating though. e.g. if the battery says max charge is 5C, that is 5 amps per 1000mah.
To be honest, if you don't have any other RC equipment, I wouldn't recommend the RC charger route unless you know what you are doing. However, this is what I do myself:
I use the stock battery with the stock charger
I use 2 RC lipos - one turnigy nano-tech 1300mah, one zippy compact 1300mah. The zippy is slightly too long for the battery tray but doesn't need any modification of the hull. Both have XT60 connectors
I use a mini-tamiya to XT60 adapter to connect to the drone.
I have an accucell 6 charger, which comes with an XT60 charging cable
I used an old laptop power supply for powering the charger at home, I happened to have one that could supply a suitable voltage and current, but I changed the plug to fit the charger. PSUs for monitors can also be suitable.