Observatories are places where astronomers do their work. Modern facilities are filled with telescopes and instruments that capture the light from distant objects. These places are scattered around the planet, and people have been building them for thousands of years. Some observatories are not even on Earth, but instead orbit or planet or the Sun in a quest for more information about the sky. However, not every such observatory has a telescope. Older ones from prehistory are simply markers that help observers capture a view of a sky objects as it rises or sets.
Before the advent of telescopes, astronomers did their observing "naked eye" from wherever they could find a dark-sky site. In most cases, mountaintops did just fine, lifting them up above the surrounding landscapes and cities.
Observatories date back to ancient times when people used rocks or sticks placed in the ground to align with the rising and setting points of the Sun and important stars. Good examples of these early ones are the Big Horn Medicine Wheel in Wyoming, the Cahokia Mounds in Illinois, and Stonehenge in England. Later on, people built temples to the Sun, Venus, and other objects. We can see the remains of many of these buildings in Chichen Itza in Mexico, the Pyramids in Egypt, and the remains of building on Machu Picchu in Peru. Each of these sites preserved a view of the heavens as a calendar. Essentially, they let their builders "use" the sky to determine the change of seasons and other important dates.
After the telescope was invented in the early 1600s, it wasn't long before people were building large ones and mounting them in buildings to protect them from the elements and support their enormous weights. Over the centuries, scientists learned to make better telescopes, outfit them with cameras and other instruments, and the serious study of the stars and planets and galaxies moved forward. Each leap in technology reaped an immediate reward: a better view of objects in the sky for astronomers to study.
Fast-forward to today's professional research facilities and we find advanced technology, Internet connectivities, and other equipment pushing huge amounts of data out to astronomers. Observatories exist for nearly every wavelength of light in the electromagnetic spectrum: from gamma rays to microwaves and beyond. Visible-light and infrared-sensitive observatories exist on high peaks throughout the world. Radio telescope dishes dot the landscapes, seeking out emissions from active galaxies, exploding stars, and more. Gamma-ray, x-ray, and ultraviolet observatories, as well as a few infrared-sensitive ones, orbit in space, where they can gather their data free of Earth's heat and atmosphere as well as humanity's tendency to spread radio signals out in all directions.
There are a great many famous observing facilities out there, including the Hubble Space Telescope, the infrared-sensitive Spitzer Space Telescope, the planet-finding Kepler Telescope, a gamma-ray explorer or two, the Chandra X-ray Observatory, and a number of solar observatories all in space. If we count the probes to the planets, plus a telescope and some instruments on the International Space Station, space is bristling with our eyes and ears on the cosmos.
The best known Earth-based observatories include the Gemini and Subaru telescopes on Mauna Kea in Hawai'i, which sit on the mountain along with the twin Keck telescopes and a slew of radio and infrared facilities. The southern hemisphere boasts the observatories of the European Southern Observatory collective, the Atacama Large-Millimeter Array radio telescopes, a collection of visible-light and radio observatories in Australia (including the telescopes at Siding Spring and Narrabri), plus telescopes in South Africa and on Antarctica. In the United States, the best-known observatories are on Kitt Peak in Arizona, the Lick, Palomar, and Mt. Wilson observatories in Southern California, and the Yerkes in Illinois. In Europe, observatories exist in France, Germany, England, and Ireland. Russia and China also have a number of institutions, as well as India and parts of the Middle East. There are too many to list here, but the sheer number testifies to the worldwide interest in astronomy.
So, can "regular people" visit an observatory? Many facilities offer tours and some give peeks through a telescope on public nights. Among the best-known public facilities is Griffith Observatory in Los Angeles, where visitors can look at the Sun during the day and look through a professional scope at night. Kitt Peak National Observatory offers public nights through much of the year, as does the Foothill Observatory in Los Altos Hills, California, Palomar Observatory (during the summer months), the University of Colorado's Sommers-Bausch facility, a select number of the telescopes on Mauna Kea in Hawai'i, and many others. There is a complete list here.
Not only can visitors get a chance to see some fascinating objects through a telescope at these places, they get a full behind-the-scenes look at how a modern observatory works. It's well worth the time and effort, and makes a wonderful family activity!