Your customers can choose which currency to view the prices on the product, collection and checkout pages in. And there's a long list of possible currencies to show. Shopify processes the order in your shop's default currency, so the shipping and payment pages only show your shops currency.
In your Shopify admin you can choose the currency for your store (Preferences > General > Money). This is the currency that your users see by default and the currency that all orders and the Shopify backend uses.
In the Awesome Theme settings you can choose which currencies to let your customers view prices in (Themes > Theme Settings > Currencies). There's a long list of currencies there, pick as many as you like.
Your customer see a currency drop-down menu in the left side-bar. When they change the currency there, it instantly, automatically updates all the prices on the screen to their chosen currency. When the customer goes to checkout, payment, and shipping, the prices are all in your shop's default currency.
Rates used by Shopify are "refreshed" twice a day and pulled from http://www.xe.com. Rates are automatically updated at 00:00 GMT and 07:00 GMT. However, it is one of a long list of jobs that are run at this time. The job goes into a queue and is completed within a couple of hours from the scheduled time.
You can read about how this works here:
http://wiki.shopify.com/How_to_Show_Multiple_Currencies
If the currency you need is not listed below, let us know and we'll add it for you.