Hi Alex,
If you have an idea of what keywords you would like to check the ranking for, then there are a couple of tools you can start using. Advanced Web Ranking has a thirty-day free trial and can handle large amounts of search volume with consistent accuracy and over 2000 search engines. This is used as an external tool and can be set to update automatically. As well, Bright Local offers a similar tool with many of the same functions, although only ten keywords can be calculated per ranking run in the free trial version.
If you do not know which keywords your site could be ranking for, trying using the Google Adwords Keyword Tool to help generate some likely ideas. As stated previously, your site could be ranking any number of unlikely, worthless keywords, and you want to measure the ones which matter.