Google saves web page versions regularly on their own servers. These pages are called the "cached version" of a webpage and are the ones Google uses for processing them and calculations, so they can answer each user request faster without having to look for the desired keywords in all the pages available online. The web pages on Google's servers allow it to find out how each webpage looked in the last saved version even when the website that published it is offline. Google just keeps a recent copy of each page.
Internet Archive keeps several copies of each web page saved throughout the years. It periodically saves web page versions found on the internet to create a log of the evolution of each web site. The above search button, looks for the latest archived web page version but it is also possible to browse previous saved versions.
Webcitation.org is a system to save web pages on demand, so authors can cite or make a reference to another page's content in their own works, and be sure that they will be available in the future. If the desired web page was referenced at some time by an article and the author saved a snapshot of the page, then It will be possible to look for it.