Las Cañitas If you are younger, adventuresome, and don't mind taking a cab to historic sites, Las Cañitas is a great alternative. The restaurant/nightlife is awesome. People who live in Las Cañitas seem to be more sophisticated; everyday you see Audis, Ferraris, BMWs. It is super safe, night and day. When you are in Las Cañitas you are closer to the big clubs (for dancing.) Pacha and Mint are two of the greatest. Mint is for a younger clientele, and more like a nightlife party. Las Cañitas has not yet been discovered as a tourism spot so renting a furnished apartment is a good idea. The Megatlon gym is within walking distance and is a very impressive place.  Beware that going downtown a taxi is expensive if they take the wrong way. Be sure they take Libertador Avenue on which it only costs 10 pesos (3 euro) to cross the city to Puerto Madero.  

Palermo Viejo and Palermo Soho One of the trendiest areas in Buenos Aires offering boutique hotels such as Bobo and HOme as well as numerous restaurants with some of the brightest young chefs in Argentina, and lots of designer shops. A beautiful neighborhood with a great mix of people, nice parks, the above mentioned excellent restaurants and coffee shops, street markets, bookstores, and boutique shops (mostly Argentinean designers.)

Recoleta Far more relaxed than the downtown area; Recoleta offers historical sights, museums, the cemetery, and many shops and restaurants, as well as bars and cafes. There are numerous tourists here and the area is more affluent than most other zones.
Many American tourists are comfortable in Recoleta as it is nice, full of trees, clean and quiet. While some think that by night it can be dark and seems dangerous, it is perhaps the single zone in town with the most visible police presence.   Indeed, this neighborhood is one of the safest in Buenos Aires. It has great sushi restaurants such as Yoko's and Moshi-Moshi as well as top quality Argentinean food at La Estancia, Beef de Lomo and Novecento. Taxis are cheap, so getting around is not a major problem.

San Telmo is also a very nice neighborhood. It is, in fact, the oldest part of the city. For many years it has been running down, but since the last few years, and mostly because of tourism, it is on the growth. Usually this neighbourhood is the tourism's favorite, as it has really charming locations, especially around Plaza Dorrego. Every Sunday there's an open air market, with tango dancers on the streets (it can get really crowded). If you are on a tight budget and looking for a hostel, you should consider San Telmo. Note that the areas around the main street (Defensa) are pretty safe, but you should be careful while wandering away from this area at night.

** Note:  The original information for this thread came from the following forum thread: