1C = 1 x Battery Capacity. i.e. a 5000mah pack has a 1C of 5A.
So a 1C charge for a 5000mah pack = 5A charge rate. 2C = 10A etc...
The high C value printed on the pack is the discharge rate. The higher the value, the higher discharge capability. 4000mah pack with a 30C discharge rate = 120A. Some packs will print the burst rating, others will print the constant discharge rating. Depends on the manufacturer.
S is the number of cells in Series. 1S is 1 cell. 2S is 2 cells in series (7.4v), 6S = 6 cells in series, etc...
2S2P pack would be 2 series, 2 parallel cells. usually you'll see this in packs with greater than 5000mah capacity. i.e. a typical 8000mah pack will be a 2S2P configuration.
The smaller set of wires with the white connector are for balacing. When you have more than 2 cells in a pack, they require balancing to keep the cells at the same voltage. You don't want the cells to drift or they can be damanged by over charge/discharge if one cell is significantly out of balance.