A "google" is the number 10^100. A "googleplex" is an even larger number 10^[(10)^100]
Google is a listed company in the USA. Their primary goal is to provide a means for people to search for information available on the Internet. Secondary goals are to provide people with tools to manage their daily tasks, like email, Internet Voice Chatting, managing pictures, etc. Aside from their usual search page they have loads of specialist search pages to help narrow down the results.
For the company Yahoo, it stands for "Yet Another Hierarchical Officious Oracle". s an American computer services company with a mission to "be the most essential global Internet service for consumers and businesses". It operates an Internet portal, the Yahoo! Directory and a host of other services including the popular Yahoo! Mail. It was founded by Stanford graduate students David Filo and Jerry Yang in January 1994 and incorporated on March 2, 1995. The company is headquartered in Sunnyvale, California.
Yahoo! started out as "Jerry's Guide to the World Wide Web" but eventually received a new moniker with the help of a dictionary. The name Yahoo is an acronym, but Filo and Yang insist they selected the name because they liked the general definition of a yahoo, as in Gulliver's Travels by Jonathan Swift: "rude, unsophisticated, uncouth." Yahoo itself first resided on Yang's student workstation, "Akebono," while the software was lodged on Filo's computer, "Konishiki"—both named after legendary sumo wrestlers. The "yet another" phrasing goes back at least to the Unix utility yacc, whose name is an acronym for "yet another compiler compiler".
Yahoo had its initial public offering on April 12, 1996, selling 2.6 million shares at $13 each.
As Yahoo's popularity has increased, so has the range of features it offers, making it a kind of one-stop shop for all the popular activities of the Internet. These now include: Yahoo! Mail, a Web-based e-mail service, an instant messaging client, a very popular mailing list service (Yahoo! Groups), online gaming and chat, various news and information portals, online shopping and auction facilities. Many of these are based at least in part on previously independent services, which Yahoo has acquired - such as the popular GeoCities free Web-hosting service, Rocketmail, and various competing mailing list providers such as eGroups. Many of these take-overs were controversial and unpopular with users of the existing services, as Yahoo often changed the relevant terms of service. An example of this would be their claiming intellectual property rights for the content on their servers, which the original companies had not done.