SEO stands for search engine optimization and search engine optimizer - a field and a person practicing the field. SEO's do SEO :)
Mentally divide SEO in two parts - onpage SEO and offpage SEO. Onpage SEO is all the things you do "at home" - on your webpages and on your site. So things like site structure optimization, content optimization and page element optimization are considered onpage SEO - you are basically doing a major cleanup on your site to make it search engine friendly.
Offpage SEO are all the activities you do off site - networking with other webmasters and bloggers with the purpose of building links through them (a link is like an endorsement of a site for search engines - the more endorsements - the more search engines "like" your site). More on SEO intro in Google SEO guide (see the sources).
You do onpage SEO first. You do offpage SEO next. You keep doing offpage SEO for as long as your site exists because link building is an ongoing process. You do onpage SEO when you have changed your website structure or added new sections to it. Though you should do both, keep in mind that offpage SEO is more important because links make up for over 50% of search engine ranking algorithm (see SEO rankings infographic listed in the sources to understand the importance of major SEO metrics).
Here is a more specific SEO workflow:
1. Keyword research - choosing SEO profitable search terms to optimize your site with
2. Onpage SEO (including keyword optimization with the above chosen keyword terms)
3. Link building - getting other sites to link to yours through networking, article writing, guest blogging, social media, contests, link bait, competitor backlink research - and a million other ways.
4. Search engine rank monitoring for your site and major competitor sites.
See SEO in Practice for full rundown on how to do SEO using this workflow with SEO tools and without. Curious to know which SEO tolls are most widely used today? Take a look at SEO software survey infographic listed in the sources.