Android - Why my phone battery has four terminal? what is the use of the fourth?

Connector A seems a U.FL-R-SMT-10 and probably is for an external/additional antenna.

U.FL, is a miniature coaxial RF connector for high-frequency signals up to 6 GHz manufactured by Hirose Electric Group in Japan.1 U.FL connectors are commonly used inside laptops and embedded systems to connect the Wi-Fi antenna to a Mini PCI card. Female U.FL connectors are not designed with reconnection in mind, and they are only rated for a few reconnects before replacement is needed. The female U.FL connectors are generally not sold separately, but rather as part of a pigtail with a high-quality 1.32 mm doubly shielded cable, which allows for a low-loss connection.

(from Wikipedia)

As @Axeman said, the connector is an Ultra Small Surface Mount Coaxial connector (quite a mouthful and still could not get how it was abbreviated as U.FL) which is a kind of expansion port which could be used for any radio communication, including GSM, WiFi, GPS.

And as @Lie Ryan said, the fourth terminal in the battery is for connecting the antennae in the battery to NFC circuitry on the motherboard. Few searches in Google confirmed this and gives raise to following information:

Those who have NFC in their phones, may lose the NFC capabilities if they try to replace with cheap and duplicate batteries which may not have the NFC antennae as said here.

I am summing up all the points to make this question as answered. I was unable to accept @Axeman's answer as complete as it did not answer the NFC part and @Lie Ryan had added as a comment only.

Kudos for them both!

Update: I happen to get hold of LG P500's service manual which confirms the RF connector with the following image tagged as SW1001.

Talking about LG, it's likely this is not any kind of NFC. I have an LG phone with 4-pin port on the phone and 4-pin battery. When I cover the 4th pin with some masking tape, the phone complains the battery is not genuine and refuses to boot picture.

A quick poke with a logic analyzer has shown me that there's a communication between phone and battery of some kind. Conclusion - it's "battery DRM", used at least on some LG phones (mine is LG Tribute). If your LG phone has 4 pin battery port, it's got this "DRM". If not, it simply doesn't care.