Zim,
Yes you can, it's the entire purpose of the API.
There is python, perl, ruby and java versions of the client library, or you can use the standard json parsing libraries in the language of your choice. The explorer should be showing you what URL you will need to hit from your code, and obviously you will need to add api_key=xxx to the url you hit to use your api key.
Don't forget to checkout the terms and conditions, essentially you must display the guardian powered by logo, and provide a link back to the original story, and you can't cache for longer than 24 hours.
Hope that helps