West Brompton is an almost entirely residential area in central London. While there are plenty of restaurants and shops, it's not a place for those looking for a wild time. Many of the residents are young families or couples just starting out. Smaller homes and apartments make up a majority of the housing stock, though there are some larger places as well.
The most famous place to hang out (and one of the only places to hang out) is the Troubadour, a bohemian restaurant with live music. There is also a tiny theater that puts on plays and musicals.
The neighborhood's Brompton Cemetery is really one of the most beautiful places in the city, as morbid as it may sound. There is an absolutely beautiful basilica and rows of graves, some of which belong to famous Londoners. If you are a literary nerd, the author Beatrix Potter took the names for the characters in Winnie the Pooh from the, er, "residents" of this cemetery.
The City: 6.7 miles / 30-60 minutes by car / 40 minutes by transit
Canary Wharf: 9.8 miles / 40-80 minutes by car / 50-60 minutes by transit
Southwark: 6.8 miles / 30-55 minutes by car / 40-50 minutes by transit
Westminster: 4 miles / 20-35 minutes by car / 25-30 minutes by transit / 25 minutes by bike
Shoreditch: 9.4 miles / 35-70 minutes by car / 50-60 minutes by transit