After being essentially leveled during World War II, the Roppongi neighborhood became home to US and Allied officials, and several foreign embassies were established here as well (the first was established about 100 years ago, but many arrived after the war). This is probably why you want to visit Roppongi now - it has a lot of nightlife, a trend re-established in the 1960s. (Roppongi was home to part of the Japanese army around 1890, and this led to several nightlife establishments opening over 100 years ago. It seems this is purely coincidental, although the choice of location for the US and Allied homes may have been based on the prior existence of Japanese military housing in the area.)
From the late '60s through the '80s, Roppongi had a lot of discos attracting foreign dignitaries and businessmen working in the area. The bubble burst of 1989 closed a lot of these places, but recent new development has led to a renaissance in the area. Today, there are lots of options, with restaurants, clubs, bars, strip clubs, cabarets, hostess clubs, and other forms of entertainment.
The area is the core of ex-pat nightlife. A secondary scene has developed in Ikebukuro and there are foreigner-owned or -based locations throughout Tokyo and Japan, but Roppongi is the center. However, this isn't all good. There have been issues with drink spiking or drink lacing, usually to minimize consumption and increase profits at all-you-can-drink bars.
Also found among the bars and clubs are some foreign restaurants, including a Cinnabon, Tony Roma's, and a few other US chains including one of two Wendy's restaurants in Japan (the other is in Shinjuku).
Also found in Roppongi are new major developments. Roppongi Hills, opened in 2003, is a city in a city, including an observation deck with great views of the city and an art museum in Mori Tower. Tokyo Midtown also is a city in a city with its own art museum, and a Ritz Carlton on the top floors. There are some other museums in the area as well, Akasaka Sacas is a little further out and the area's newest city in a city, with great dining but fewer attractions.
If you're looking for foreign-friendly nightlife options, Roppongi is the place to go, and it's probably where you'll see the most expats in Tokyo (instead of just foreign tourists). That said, the area doesn't have a big repeat draw for me since I'm not much for club hopping, although Mori Tower's art museum frequently has good exhibits and sometimes a Wendy's spicy chicken sandwich really hits the spot.