The Bay Area is The San Francisco Bay Area.
It begins at the Pacific, stretching from Marin County across the Golden Gate Bridge in the north, Santa Cruz to the south, with the City of San Francisco and the Peninsula in-between.
It includes the South Bay, also known as Silicon Valley (note: “Silicone” – aka San Fernando – Valley is a bit further south).
And of course, the East Bay.
That's where Oakland is.
Now you know :-)