The paid adverts are on the right hand side and often right at the top of the results in a coloured area to show you that these are paid ads. To appear here you have to pay and if a user clicks on the link to your website you will pay for the click and this can become very expensive.
Depending upon the budget of the Adwords campaign your website link will appear as long as the budget or campaign settings allow this.
Organic listings are there 24 x 7 and do not cost if someone clicks on the link, so in the long term being in the organic listings is much better. To achieve organic success you need to carry out Search Engine Optimisation.
What is Search Engine Optimisation
Search engine optimisation is a complicated process where an SEO consultant will look at your website and understand what keywords you should being going after and then making the search engine find the information on your website to rank the website for these keywords. A search engine will return the best results for a user based on the keyword that has been searched and the SEO consultants job is make your website attractive to search engine.