Search engine results pages or SERPs are the listings of web pages returned by a search engine in response to a keyword/phrase query. The results normally include a list of web pages with titles, a link to the page, and a short description showing where the keywords have matched content within the page. A SERP may refer to a single page of links returned, or to the set of all links returned for a search query.
The list of websites consists of two different types of results:
The websites appearing under the sponsored listing section which is normally at the top of the page and to the right, have all paid to be there using Pay Per Click campaigns. This is called paid inclusion.
The rest of the results are natural or organic search engine results. Your site needs to be well optimised using SEO techniques to appear high in these natural listings.
Why do I see different SERPs when searching more than once?
There are many data centers or servers which all day go around indexing pages. Each server will have similar but not the same results. So whenever you search on Google, Google will go to the quickest route to get your query answered. This could mean using a server in another country if it means getting the result fastest. If the closest server will get results in say 0.5 seconds, but a server further away may be able to deliver results in say 0.3 seconds, it will show results from the quickest server. So it all depends on the amount of workload a server has.
Also if your website is not mobile friendly it might show up on a desktop pc but not on a mobile device.