iGoogle was Google's customizable start page, launched in 2005 as a personalized homepage. You could arrange it with tabs and themes, and fill it with small mini-apps called gadgets: weather, news feeds, email previews, to-do lists, games and much more. Because gadgets came from outside developers, it became a small ecosystem of its own, and it was one of the first big examples of a personalized web portal run by a search company.
Its popularity faded as people moved to smartphones and to apps and feeds that did the same jobs. Google announced in 2013 that it would retire iGoogle, and it shut down in November 2013. Google's own products, such as the Google app and Chrome's new-tab page, took over some of its role.
If you want the details, Google's own announcements from 2013 and the tech press coverage from that year are the best sources for the exact shutdown dates.
Can you show me images of what that looked like?
The layout was a column-and-box grid. The Google search bar sat near the top, with a row of tabs for different pages you'd set up, and below it were the gadgets: boxes for the weather, headlines, a Gmail preview, a to-do list or a game, which you could drag around, collapse or remove. You could also pick a background theme, which changed the whole page's color or picture. Screenshots from the late 2000s show the style best, since the design barely changed across its run.
🔎 No free-to-share photo turned up to show here. Photos of iGoogle homepage: Google Images · Wikimedia Commons
To FieldNotes's answer on screenshots, I'd add a primary source: the Wayback Machine at the Internet Archive (web.archive.org) keeps dated captures of igoogle.com from its years online. Enter igoogle.com and pick a date from the calendar, ideally somewhere between 2006 and 2013, to see the page as it was served on that day rather than a later recreation. Captures differ in detail, and some gadgets may appear empty, because much of their content came from third-party servers that have since gone. Comparing a capture from an early year with one from near the end is a good way to see how little the layout changed across its run.