You are standing in Union Square, so called because this is where Thomas Starr King led the pro-Union rallies, urging Californians to support the North. His church faced the square at the time, although it has since moved.
This is one of the first squares in the city, land for it having been gifted by Colonel Geary, the first American mayor, in 1850. At the time it was a large sand dune, covered with gold-diggers' tents. Later it was leveled, and became a favorite place to play baseball. In fact, this is where the term "sand-lot baseball" originates.
Underneath your feet is the nation's first underground parking garage, built in 1939 by Timothy Pflueger.
Above your head is the Goddess of Victory, atop of the Dewey monument by local sculptor Robert Ingersoll Aitken.
The trident and laurel wreath she holds symbolize the victory of US fleet led by George Dewey (USA's only Admiral of the Navy) over the Spanish fleet in the battle of Manila Bay. US troops were deployed from the Presidio, which is why this war was more special to San Francisco than to other cities.
This is the first monument built in USA to celebate one of the most decisive battles in naval history. USA lost one man (Francis B. Randall, to a heart attack). Spain lost 381 sailors, their entire Pacific fleet, Guam, Philippines, Cuba, and status as a colonial power.
The Filippino-American War that resulted from the US victory in the Spanish-American War was just as horrifyingly brutal as you'd expect, perhaps even a little more so. But we're talking specifically about San Francisco, so I can skip right past General Jacob H. Smith (seriously - don't google him) and just tell you that today there's a large Filipino community in San Francisco that thrives despite a history of adversity caused by prejudice. Their annual Pistahan is the largest celebration of Filipino culture in the US.
The general consensus is that Nike, the Goddess of Victory, was modeled from Anna de Bretteville in her full six-foot-plus glory, but the Chronicle, in 1902, claimed the model to have been Clara Petzold. I prefer the former version.