Outside of Asia, San Francisco has the largest Asian population in the world! The result is an authentically Chinese experience when you visit Chinatown.
However, there are really two faces to this neighborhood. On Grant Street, souvenirs and restaurants with English translations on the menus abound. This is the tourist version of Chinatown and though it's fun, it isn't a true representation of the Chinese community. If you step one block west to Stockton St., your experience will be quite different.
On Stockton, the sights, sounds, and smells are an authentic glimpse into the "real" Chinatown. Be prepared for crowded streets and lots of foodstuffs you've never seen before! Though there are some shops on Stockton, the focus is on groceries and restaurants. Look for restaurants offering Dim Sum, where carts of different appetizer-type concoctions come by your table. You're charged for each plate you accept. It's a great way to get familiar with the uniquely Chinese fare.
You can easily walk to Chinatown from Union Square. Though it's within walking distance from Fisherman's Wharf, as well, the distance may be slightly more than some travelers are willing to go. Public transportation (MUNI) serves the area well. Chinatown is next door to North Beach.