Organic Search Engine Rankings refers to where your page appears when someone types in keywords in a search engine such as Google. Your site needs to be optimized to get the best possible ranking.
Your typical Shopify theme is already compatible with the technical aspects of SEO (Search Engine Optimization) - following the design guidelines that make it easy for search engines to index your content, but first the Search Engines have to know about your site...
Submitting your Site
To get started, you need to submit your site to the search engine. Here are some links:
Once you have submitted your site it is added to the queue so that their "spiders" will index it which will normally take a few days.
Tuning your Keywords
While you are waiting, you might want to spend a few minutes tuning your content to your specific keywords (you have specific keywords right?). You can get an idea of what the search engines are seeing by using this free tool: http://www.seomoz.org/term-extractor. It will analyze the content of your page and extracts the terms that the search engines are seeing. Hint: it doesn't hurt to run this tool over your competitors' sites as well, to see what they are using.
High quality content, tuned to your keywords, will be the single most important factor in improving your organic search results.
Using Paid Search
Did you know you can also use paid search (i.e Adwords "PPC") to help your organic rankings? Adwords gives you a quality score for your landing pages based on the keywords used. An initial quality score will be around 4-5 if you have decent content. If you then follow the Adwords help on improving the quality score to say at least a 7 you will find your organic ranking will improve as well.
Another critical factor in your organic rankings is how many sites are linking to you. Search engines consider links as votes, but not all links are equal. An "authoritative" site (one that has a lot of links itself) will have a higher value.
The last thing I'd like to mention is that Search Engines factor in the age of domains. In other words, they have to be around for a while so you will have to be patient.
While we are on the topic, you should know that the GoogleBot loves to see fresh content as well, which is why blogs are so effective.