The Internet Archive (link) was started in 1996 to catalog and preserve the web. Today it has saved 435 billion webpages and makes this massive historical collection easily searchable. the site is constantly updated with new snapshots and every couple of years the sites platform is upgraded to keep is modern and working.
For those of us that have been on the web from very early on, there is something heartwarming and nostalgic seeing the early versions of Geocities, AOL, Compuserve and even Facebook.
There is also a business use to this site and it has to do with research and brand protection. Research because academics can use it to analyze historical changes to the web. The brand protection aspect comes in because you can check what the domain name was used for before you bought it (porn site, medication sales, SPAM site, etc).
If you bought a site that Google considered questionable and was therefore blocked by the search giant, you can use their reconsideration form (link).
So go and enjoy the web as it was and think of what the next generation will think of our current sites.