In the 20th century, Brooklyn was an immigrant haven and a beloved hometown to millions before suffering a severe decline after the middle class fled to the suburbs.
In the 21st century, it is experiencing a renaissance perhaps unparalleled in modern urban history. To reduce it to a couple of telltale statistics: Crime has dropped 75 percent since the 1980s; the population, which plummeted by more than 465,000 from 1940 to 1980, has increased by about 200,000 since 1990. 
But the bigger news may be Brooklyn's emergence as a tourist destination, for more than just an afternoon excursion from Manhattan, maybe even for a vacation all its own. You now have a choice of accommodations in Brooklyn ranging from chain hotels to boutique properties to B&Bs.
Brooklyn has always possessed one of the great tourist attractions—and architectural wonders—of the world: the Brooklyn Bridge, with its beauteous Gothic stone towers and swooping mesh of steel cables. The wood-planked, pedestrian-only upper level of the bridge provides an awe-inspiring view of the New York Harbor, Manhattan's skyscrapers and the Brooklyn Bridge's neighboring East River spans, the Manhattan and Williamsburg bridges (which also have pedestrian lanes). And there's no admission fee!
Get on the bridge opposite City Hall in lower Manhattan. When you complete the half-hour walk across, you can continue on into one of three neighborhoods. Go to your right and you'll enter Brooklyn Heights, a splendidly preserved enclave of 19th-century brownstones and mansions, with a Promenade overlooking the water. Proceed straight ahead off the bridge and you're in Downtown, with such historic and architecturally impressive buildings as Borough Hall (built as Brooklyn's City Hall in 1849), plus the New York Transit Museum inside a decommissioned subway station. 
Make a 180-degree turn when you get off the bridge and head into Dumbo (for "down under the Manhattan Bridge overpass"), a former industrial hub whose old brick warehouses are now filled with luxury apartments, art galleries, boutiques and eateries. It's a cozy area, with cobblestone streets, a riverfront park and the old Ferry Landing (get ready to ooh and aah at the views).
Culture and Nature
Brooklyn spans some 80 square miles—versus Manhattan's 22—so there's a lot to see farther away from the bridge. Fort Greene, on the other side of Downtown, has some of Brooklyn's loveliest brownstone blocks, many of them bordering hilly, verdant Fort Greene Park. Also located in Fort Greene is the Brooklyn Academy of Music—better known by the one-syllable acronym BAM. This is the United States' oldest performing arts center, originating around 1860 and occupying a stunning Fort Greene building embossed with cherub musicians and multicolored terracotta ornamentation.
When visiting Fort Greene, don't miss Clinton Hill, the adjacent neighborhood to the east. Walk along Washington and Clinton Avenues to see the 19th-century mansions built by moguls of the day and wander through the sculpture garden that's actually the campus of Pratt Institute, a college founded by one of those moguls, oilman Charles Pratt.
Brooklyn's cultural center begins at Grand Army Plaza, with its magnificent triumphal arch. Walk along Eastern Parkway—a Parisian-like boulevard with landscaped pedestrian byways flanking the road—to the 52-acre, 13,000-species Brooklyn Botanic Garden and, just beyond it, the Brooklyn Museum. Housed in a gorgeous Beaux Arts building with "dancing waters" out front, the museum features world-class collections of Egyptian antiquities, European, American and African art, furnished period rooms and a brand-new Center for Feminist Art.
Bordering Grand Army Plaza on another side is Prospect Park, a 500-plus-acre expanse of rolling hills, waterways and natural woodlands. Within the park are a zoo, carousel, Palladian boathouse and circa-1780 Dutch farmhouse.
To its west is yuppie Park Slope, which boasts street after street of beautiful brownstone and limestone rowhouses, as well as some elegant old mansions. Like the Heights and Fort Greene, it's full of historic churches bound to catch your eye.
Venture farther out to Coney Island, the famous beach destination with its boardwalk, rides galore, Nathan's (for hot dogs and fries), an old-fashioned freak show, the Coney Island Museum (a trip down memory lane well worth its 99-cent admission) and the excellent New York Aquarium.
Coney Island is not the only place in Brooklyn to enjoy the sea air. Out near Jamaica Bay, you can traipse through Marine Park's Everglades-esque salt marsh, home to a mile-long nature trail and hundreds of bird species. In Bay Ridge—where Saturday Night Fever was set and filmed—a pedestrian esplanade runs alongside New York Bay from the pier at 69th Street (another awesome view of lower Manhattan and the Statue of Liberty) to right under the Verrazano-Narrows Bridge.
And how unhip am I that it took me till the end of the story to mention Williamsburg? You can go for the nightlife—an exciting variety of restaurants, bars and offbeat performance spaces—or spend a day wandering in and out of small art galleries and coffeehouses. On Saturdays, the Brooklyn Brewery offers free tours, or go there on Friday evenings to drink what's produced onsite. At the foot of the Williamsburg Bridge, there are a number of historic structures, including a grand domed bank building erected around 1870.