It is important to note that just because you might offer visitors to your website the chance to pay for your products in different currencies does not mean that your website serves multiple countries. Unless you want to use multiple languages or have different product/service offerings for different counties, you can carry on having one website in your primary language and simply let users view your prices in different currencies or just check-out in different currencies.
If your business offers different products or services to people in different countries, and in particular if you plan on using multiple languages, the most appropriate approach is to have a separate website for each country you want to serve. Each website should feature the local domain extension and use the primary local language (for example "Amazon.com", "Amazon.co.uk", "Amazon.fr" and more), which makes sense because you are offering a customized product/service range for that country and because websites that have content written in a language that matches the domain extension (such as French content on Amazon.fr) will tend to rank higher for searches in that language.
How you set-up each of these websites relative to each other is largely unimportant, because they will in fact be completely different websites, with different content. Google's Gary Illyes suggests that you suggest more appropriate language versions to users based on their location - for example if you access the .fr version from Germany you could display a prominent option to visit the .de version instead.
If your business offers the same products and services to people in every country, but you want to specifically target users in different languages, the most common approach is to maintain just one website but have a variety of language options. You can either let your visitors select their preferred language when they arrive on your homepage, or you can automatically redirect them or display country-specific content based on their location. It is important to note that it can be irritating to be automatically redirected to different language versions, for example for Americans on holiday in Europe, so it's important to allow visitors to be able to override such automation if they so choose.
When creating a website with multiple languages, there are certain steps you need to take to make sure that search engines properly understand the structure of your website and your rankings are not harmed:
As we have seen, while catering to multiple countries and languages can be complicated to implement correctly, it is also sometimes a necessary step in reaching new audiences and new customers. If you make a lot of sales in currencies other than your own, or are planning to launch new products or services that are tailored to specific countries, it may be time to consider organizing your web presence accordingly.