On April 22, 1964, New York opened its most space age World's Fair. The centerpiece was a 120-foot diameter sculpture called The Unisphere, the largest globe-style sculpture in the world. Made from stainless steel and standing twelve stories tall, the Unisphere was meant to evoke the fair's theme of "Peace through Understanding." It still stands today in Queens.
The three rings surrounding the Unisphere symbolize the orbits of three great firsts in space: Yuri Gagarin, the first human in orbit; John Glenn, the first American in orbit; and Telstar, the first communications satellite.
Oddly, the Unisphere is actually the second World's Fair-related sphere to be built in Flushing Meadows Corona Park. In 1939, the Perisphere was built in the park for that year's World's Fair. It was later scrapped.
To get a sense of how awesome the Unisphere was, check out this 1964 commercial advertising the New York subway:
For more on the experience of the fair, the Unisphere, and what's left today (lots of sculptures!), check out this delightful tour by Michael D. Jackson, mixing historical and modern footage:
The Unisphere and the New York State Pavilion remained after the fair closed, but fell into disrepair. The band They Might Be Giants explored the area in their songs (they discuss the "'64 World's Fair" in "Ana Ng" among other songs), and even shot the video for "Don't Let's Start" in the pavilion in 1987. The grounds were restored starting in 1989, and appeared in a variety of movies, including Men in Black, The Wiz, and Iron Man 2.
Today you can visit the Unisphere easily. Just take the #7 train to 111th Street and walk a few blocks. The fountain is only turned on in the summer, but you can visit any time. (Flushing Meadows Corona Park also contains a bunch of other stuff.)