Paid search advertising is a marketing method where you pay for your website to appear in the paid advertising sections of relevant search engine results pages. By selecting terms that your ideal customers would search for when using search engines, you can create adverts that appear on search engine results pages based on those search terms.
Obviously, there will be competition for the limited advertising spots allotted per search engine results page for search terms. Advertisers accounts are set up to bid against each other for how much they are willing to pay every time a search engine user clicks on an ad. The higher the bid in pay-per-click (PPC) advertising and how relevant the ad is to searchers, the more likely it is to appear in search results.
Paid search advertising is big business ($52 billion for Google alone in 2015), so expect it to be around for a while. Businesses struggling with ranking high organically with SEO or new websites benefit from using paid search advertising to gain exposure to a relevant audience and get seen on page one of search results.
Google provides paid search advertising through AdWords, Bing has Bing Ads, and Yahoo has Yahoo Gemini. This is how Google, and other search engines, make a majority of their revenue.