I've made the decision to, at least for now, open up the API to anyone who wants to use it without authentication. This means that using the service is as simple as fetching the response of a URI.
To get basic information for all of the countries in the world, go to http://geodata.solutions/restapi
If you want a little more info on Canada (and let's face it, who wouldn't?), it's http://geodata.solutions/restapi?country=Canada&limit=10
If it was British Columbia that really peaked your interest, go to http://geodata.solutions/restapi?country=Canada&state=British+Columbia
All of this is made easy using the API wizard, where you can choose what information you want, see a preview of the output, and be provided the URI.
Allowing for open, unauthenticated access seemed like the right thing to do in order to get people using the system quickly, and to grow in popularity and features. Over coming week, the intention is to produce more focussed API queries that are suited for specific tasks, such as jQuery code examples for Country -> State dropdown sets that might allow users to fine-tune their operation to suit their requirements. Other implementations might be getting latitude and longitude data for mapping purposes, fetching the Japanese names for countries, comparing the geographical size of countries, ordering lists by timezones, creating population heat maps for the USA, or various other things.
I encourage you to try out the API. I've worked hard on it. See what uses you can find for it, and get in touch if you have any specific requests in mind.