An SEO company is a company that carries out Search Engine Optimisation on your site with the main goal of improving its search position in search engines such as Google, Bing and Yahoo. This boost in search engine visibility results in more traffic to your website meaning more sales, leads and calls.
There are quite a few ways that an SEO Company can achieve this. Some methods are considered against Google guidelines (Black Hat) while others are perfectly within Google's guidelines (White Hat).
Usually SEO involves creating a better user experience on the site, making sure search engines are able to easily decipher what the page is about and building relationships and links with other sites in your niche establishing your website in an influential and authoritative position amongst your competitors.