There are many vertical, needs, purpose for SEO Analysis. As @Mellisa mentioned there could be just these areas as well if a site has been penalized then identification of Penalty, link audit, website audit.
Mostly it depends what is state of website, is it absolutely new or was doing good and suddenly disappeared, or there is need to start optimization.
During analysis you may need various tools for analysis, Here is a post - Tools, checklists to create high quality website
From here you may get lots of tools which you might need for website analysis