I think that by now most of us are aware of the Wayback Machine at archive.org, but just in case you're not, it's a service that has been around for well over a decade, and it archives websites and pages. It allows you to browse through URLs that were produced from 1996 up to December 9th 2012. Not every single site or page has been indexed (crawls can miss them, and owners can request that their sites not be included), but rather a lot are. Recently the index has been updated and has gone from having 150,000,000,000 URLs to having 240,000,000,000 URLs, a total of about 5 petabytes of data. This database is queried over 1,000 times a second by over 500,000 people a day helping make archive.org the 250th most popular website.
It's extremely easy to use - just type in the address of the site that you want to check out the archive for, and you'll see a series of dates going back to - in theory - 1996, or whenever the site started. You can then choose your archived time, click on the link and view the site as it existed then. You won't find the entire site, and sometimes you'll find images missing, but it'll give you a fair indication of the content of the site.
There are as many different ways to use the Archive as there are users, but some basic ideas are as follows:
Check to see the history of a site - has it always been owned by the same person/company, or has it changed hands over the years?
Find out information about a company that's not currently on their site - who was the CEO five years ago for example.
Compare attitudes towards different subjects then and now. Has the position of a person or company materially changed over a period of time?
Initial responses to events - what was the first statement that a person or organisation put out publically about some event?
How webpage design has changed over the years/months.
View world events, such as 9/11, or national elections.
Check out early pioneers of the internet - who was doing what when.
Compare previous versions of search engines and their functionality with what they can do today.
Cite specific examples of webpages, even when they have changed.
Content stored in the Archive has been used in legal cases.
Find archives for a site that has changed URLs.
Locate a manual for an obsolete piece of kit.
View a site that has been suspended for whatever reason, or which you cannot reach.
Compare old/new prices.
Find old predictions and see how accurate or not they have turned out to be.
That's a quick 15 examples, but there are plenty more valuable ways of using it. The Wayback Machine should be a really useful weapon in any information professionals arsenal.