It is all to easy to have a website designed and not to keep track of the progress it is making in terms of popularity on the search engines and visitors.
A site needs to be designed in such a way that it performs well on the search engines and also is user friendly as having one without the other simply will not work. To analysis this you can use statistics programs, Web Master Tools and other data that you collect yourselves. Ideally you want to see an improvement in terms of where you rank for your main phrases and would also want to see the number of pages per view and time spent on the site increase. Obviously if it is an e-commerce site you would also hope to see an improvement in sales or enquiries for any other site.
Monitor the website on a monthly if not weekly basis and make a note of any changes you make to see how they affect the visibility and effectiveness of the site.