The Bay Area (more fully, the San Francisco Bay Area), ringing the San Francisco Bay in northern California, is a geographically diverse and extensive metropolitan region that is home to over 7 million inhabitants in cities such as San Francisco, Oakland, and San Jose. Once a focus of Spanish missions and Gold Rush prospectors, the Bay Area is best known now for its lifestyle, liberal politics, and high-tech industry (Silicon Valley).
Although it doesn't have any firm boundaries, the Bay Area includes portions of nine counties: Marin, Sonoma, Napa, Solano, San Francisco, San Mateo, Contra Costa, Alameda, and Santa Clara. San Mateo and the northern part of Santa Clara counties comprise the Peninsula while Contra Costa and Alameda counties comprise the East Bay of the Bay Area. The southern parts of Sonoma and Napa counties are considered part of the Bay Area for this guide since their culture and economies face towards the Bay. To the East and South, whether outlying towns such as Gilroy and Santa Cruz are part of the Bay Area or the Central Valley/Central Coast respectively will depend on who you ask.
Here's a handy rule of thumb: the telephone area code 415 generally means San Francisco (although it includes a few nearer parts of the North Bay like Marin); 650 is the Peninsula, 510 and 925 is for the East Bay, 707 for the North Bay or Wine Country and 408 and 669 are for the South Bay or the Santa Clara Valley.
Note that the boundary between Peninsula and South Bay is not formal; some locals and some other sites/guides may place some of the northern Santa Clara County cities categorized here as "Peninsula" as South Bay - for example, the local Craigslist does so for Mountain View.