Search Engine Optimization (SEO), is the process of getting "organic" traffic to your website. ​

Organic, or "real" listings will only show up in search results by being a relevant answer to a search.  
Just having a website doesn't mean that your potential customers will find it. Google, the most used search engine is in the answers business.  Google wants to provide the most relevant answers to "questions", aka "searches".  
In order for your website to rank high in search results, it has to be seen as a relevant answer by Google.  
There are guidelines that you must follow to become a relevant answer to a search.  SEO programs work toward making your website a relevant answer, thus ranking it higher in search results.  
SEO is a process, not a quick fix for low rankings.  You cannot buy a higher ranking, it is a process that takes time.  For a quick fix, you can use Search Engine Marketing.

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Search Engine Marketing (SEM), is the "instant" way for your site to show up in search results.  

With SEM, you can control your message.  You set your budget, and you only pay when someone clicks on your ad.  It is easily trackable, and it helps with your SEO by driving traffic to your site, which makes your site more popular.  Popularity is one of the factors in the way Google ranks your site.

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Example of SEO (organic results) & SEM (top 3 listings) in a Google Search result: