As you stated, gold/silver/copper is the standard generic currency. It is used to buy/sell things on the auction house, repair item, buy gems, & buy things from general merchants. It is earned by killing stuff, selling monster drops, quests, opening loot bags, & from events.
Gems are the premium real money currency. It can be bought with either real money or with gold/silver/copper. It is used to buy cosmetic items and to upgrade some account/character features. It is not a 'pay to win' currency. If anything it is a 'pay for convenience' currency.
Karma is awarded by completed events and quests. It is spent at karma vendors. Theses are special shops that sell some specific items (crafting materials and magic items) and only take karma as currency.
There are also some other forms of currency whos, names are currently escaping me. One that is rewarded as part of PvP and one that is rewarded from kills in WvW. Each of them are used inside of shops within their respective zones.