The below explanations applies to search engines in general. But for the sake of simplicity and clarity, we are going to use Google as an example (since it is the leading search engine at the time of this writing).
Let's say you have a website about catnip. You want your site to be found when people search for "catnip" in Google. Well, so does many other catnip site owner who also wants Google to list them first when "catnip" is searched. But what determines which site Google put on top in the "search engine results page" (also known as SERP)? This is determined by literally dozens of factors and only Google know that exact algorithm and weightings of each of these factors. It is general agreed that these factors includes things like the text on your page, the title of your page, how many other sites are linked to your site, your URL of your page, how long you have your domain name, and many other factors. These factors can be loosely categorized as internal factors and external factors. The internal factors are elements on the page itself such as the text and the code that you can control. Hence, they are also known as "on-page factors". The external factors (also known as off-page factors) are factors such as the quantity and quality of the links to your site and the length of ownership of your domain name, etc.
The process of trying to optimize all these various factor in the attempt to have Google list your site on top of the search engine results page is known as "search engine optimization", or SEO for short. Consultants whom you hire to perform this optimization for you are known as "SEO consultants". And they may do things like suggest different text copy or alter your page's code as well as "link building" (going out to relevant websites owners and asking them to link to your site).
Due to complexity of all these factors and the amount of competition from other sites, search engine optimization can be difficult and there is no guarantee of results for top search placement.
So far, what we have been referring to above is "organic search results". These are main search results that you see on the main column content of the search results page. And its list order is solely determined by how relevant the site is in the eyes of Google to the keyword that the user is searching for.
However, notice that Google search result page has a right column that lists ads known as Google search results ads. These are sponsored search results listing -- meaning that merchants pay Google in order to be listed in that right column. The merchant pays Google a small amount every time the merchant's ads is clicked by an user. The order of this ads in this sponsored listing is based on how much the merchant is willing to pay Google for each click. This payment scheme is known as "cost per click" (or CPC for short). Whether your ad shows on a particular search results page is determined by the keywords selected by the merchart when setting up the advertisment campaign.
A merchant who would like to participate in this Google's advertisment campaign may do so by signing up with Google AdWords. This is known as "search advertisement" and here is a video of some success stories. And you can watch free training videos at Google's Adwords Online Classroom.
