SEO is an acronym for Search Engine Optimization.
A "search engine" is a service like Google, Bing, Yahoo etc. The phrase search engine is an 80's phrase. No one uses the phrase in everyday language. Search engine refers to the algorithms that Google, Bing or Yahoo employs. Google has become the standard search engine. People don't say, search for it, they now say Google it.
Here is the wiki definition SEO
SEO is a service, an event and a process. This service involves someone editing and adding content to your website, that encourages Google to rank your website as important and worth visiting.
Believe it or not, Google examines your website regularly, frequencies ranging from daily to every week. Google compares your website to others, in your industry and then gives it a ranking typically 1-200 (or worse). When someone searches for your service, Google will list you according to the ranking previously determined.
Unfortunately, a beautiful website, is often the best kept secret on the internet. Google does consider how much time people spend on your website, but if they don't find your website at all, then having a beautiful website doesn't really matter.
SEO services (Search engine optimization services) is not something that happens on its own, or happens when you do a new website, it is an intentional process, not related to website design.
For an SEO professional it should take over 100 hours effort. For the average everyday person, add another 400 to 600 hours of learning and trial and error.
When engaging an SEO professional, your biggest leverage is your money. If the SEO provider is asking for the full payment upfront, be wary. Arrange a payment program that spans the entire duration of the SEO engagement.