There are two airports in Buenos Aires, Ezeiza International (called Aeropuerto Internacional Ministro Pistarini) and the domestic Jorge Newbery Airport (called Aeroparque). They are fairly close to the city, about a half hour drive by highways.
As Argentina has been a tourist hub for some time, you will find that the variety of facilities available is quite good and there will be people or services to cater for almost anything you need. This also means however, that there are plenty of tourist traps and locals trying to get the best of unsuspecting travelers. This is especially true with taxis. Be wary and if in doubt, spend a little extra for safety and security.
There are coaches that run fairly frequently from the airport to Retiro, the main station in the city. These are clean and safe and tickets cost about 50 pesos from a ticket booth just outside the customs area. Look out for a Manuel Tienda León sign. The same company also has coaches that run to the domestic airport, Aeroparque. Website: http://www.tiendaleon.com.ar/prehome.asp
Public buses run through the Ezeiza (international) Airport as well. The number 8 bus stops outside terminal B arrivals and takes about 2 hours to get into the city. If you are familiar with the roads of Buenos Aires, this can be very cheap as a ticket is about 2 pesos. Exact change is normally required. Of course, these buses can be crowded and take a long time to get anywhere, so chose your mode of transportation carefully.
Remises, or prepaid taxis are the general method of transportation from the airport, especially if you are laden with much luggage. About 150 pesos will get you anywhere in downtown and they can be booked online as well.
If you are familiar with the Buenos Aires area and speak Spanish fluently, a non pre-paid taxi is also an option. To make sure they are genuine, get into one that is dropping someone off rather than one which is waiting. This could be a cheap option with a trip to town costing about 55 pesos. Road tolls will get added to your fare.
Private drivers will also vie for your attention, they may be expensive but they will get you exactly where you want to be in comfort. Most speak English.