Meat

You will want to try asado (beef/steak barbecue) at a parrilla, restaurants specializing in roasted meats. There are expensive parrillas, and more simple and cost effective ones, . The bife de lomo (tenderloin) is unbelievably tender. Jugoso means rare (literally "juicy"), however the Argentine concept of rare is very different from that of someone from the States (perhaps its a tourist thing, but an American ordering rare is likely to get something between medium well and hockey puck). Don't be afraid to order "azul" (blue), you will not get a blue steak, more like an American Medium Rare. If you like your meat bloody it might pay to learn words like "sangre" (blood), or to make statements like "me gusta sangre" (I like blood). Don't be afraid to spend two minutes stressing how rare you want your steak to your waiter- this is something no one talks about in guidebooks but every other American and Brit once you arrive will tell you the same thing.

Vegetarian

In Buenos Aires, and in the rest of the country, beef is king, but it's not your only option in this cosmopolitan city. Italian food is pervasive but in neighborhoods like Palermo, pizza joints are seeing heavy competition from sushi, fusion, and even vegetarian bistros. Just about everything can be delivered - including fantastic, gourmet helado (ice cream).
Vegan food is available at these restaurants:
Artemisia - 3877 Cabrera
Bio Restaurant - Humboldt 2199
Bodhi - Chile 1763
Granix - Florida 165 2nd floor
Green Life - Paraguay 2743
Los Sabios - Corrientes 3733
Lotus - Cordoba 1577
Prana Cocina Vegetariana - El Salvador 5101
Sattva - Montevideo 446
Siempre Verde - Arribeños 2127

Pizza

Italian and Spanish food are almost native here, as the cultural heritage heralds in great part from these two countries. Other popular meals are pizzas and empanadas (small pastries stuffed with a combination of cheese and meats). They are a popular home delivery or takeaway/takeout option.

The pizza is excellent in Buenos Aires, due to the Italian immigrant heritage. Pizza comes al molde (cooked in a pan, usually medium to thick crust), a la piedra (baked in a stone oven, usually thin to medium crust), and a la parilla (cooked on a parilla grill, very thin, crispy crust). Best places: "Los Inmortales", "Las Cuartetas", "Guerrín", "Banchero's".

Sweets

One incredible and typical Argentinian kind of "cookie", is the alfajor , which consists of two round sweet biscuits joined together with a sweet jam, generally dulce de leche (milk jam, akin to caramel), covered with chocolate, meringue or something similarly sweet.
Service: do not expect service to be comprable to large cities in Europe or in the USA. Service in Buenos Aires can be slow at best and horrific at worst. Don't expect your waiter to take your drinks order when the menu is delivered and don't expect the menu to arrive quickly. If you want ice in your drink, expect the drink to arrive several minutes before the ice does. If you want service, attract the waiters' attention, s/he will never come over to take your empty plate etc. unless they want to close.
Patience is the key. Argentinians as so accustomed to bad service that they don't bother to complain direclty to the waiter/waitress but moan amongst themselves. Speak out if you feel it is appropriate.