You have about half of it right. Club Nintendo rewards your new Nintendo-related purchases -- most hardware, most games published by Nintendo, usually not games on Nintendo consoles published by third parties, with some exceptions -- with virtual "coins." You can exchange those coins for certain rewards.
If Fire Emblem: Awakening is "worth" 60 coins, that means you will receive 60 coins if you buy it and register it on Club Nintendo. It doesn't mean you can buy the game for 60 coins! Usually the games on offer are Virtual Console or WiiWare/eShop titles, and they usually cost 150 or more coins apiece. For instance, right now you can "buy" Super Metroid on the Wii Virtual Console for 150 coins, or Super Mario 64 on the Wii Virtual Console for 200 coins.
Also, if you register 300 or more coins (doesn't matter whether you spend them) throughout the year, you'll qualify for a gift from Nintendo at the end of June. Last year, I got Metroid II on 3DS Virtual Console through that program, but there were other options as well.
