Google Analytics is not Search Engine Optimization. It is a free analytical tool you can get from Google once you register. Each website you have will be given its own unique registration number. You have to copy and paste a snippet of code into your website for Google Analytics to start working. Once active, you can track all sorts of activity on your website.
You can track user activity by visitors, location in the world, they amount of time they spend on your website, your bounce rate (number of people who enter and leave on the same page without doing anything). You can get goals and monitor your conversion rate on the goals. A goal can be defined by you. It may be downloading a PDF white paper r, in the case of an e-commerce site, clicking the 'Buy Now' button. You can define these individual goals and then monitor the number of times they have been completed.
Another neat thing you can do with GA is you can benchmark your site against similar sites in the same industry. For example, if the you are a Florist selling on line and your bounce rate is 62% when the industry bounce rate is 45%, you have some work to do on your site. How does this help you? It lets you see an abundance of statistics about how uses interact with your website. From the Google Analytics tools available, you can better understand and analyze how your site is working. More importantly, how it is not working.
Using Google Analytics will not advance your ranking placement in Google. It is simply an analytical tool. In order to do this you need a well-executed SEO plan. Focusing on the keywords in your web content is key here (not the keyword meta tag, as Google doesn't use this). There are other critical tags contained within your web page that do count highly with Google and make the difference in ranking better.