The national drink is called Airag. It is available in for example in traditional mongolian "ger" tents in Ulan Bator at the main entrance of Gandantegchinlen Monastery, GPS decimal coordinates N47.92069 E106.89467 for 1500T and a the West Market N47.91118 E106.83569 for 1000T per bowl as of September 2010 This is a summer seasonal drink made from fermented mare's milk, and is certainly an acquired taste. The alcohol content is less than that of beer, but can have noticeable effects. Be careful, if you aren't accustomed to drinking sour milk products the first time might give you diarrhea as your stomach gets accustomed to it. This should only happen the first time though. Once you've completed the ritual, your digestive system shouldn't complain again. There are numerous ways to describe the taste, from bile-like to a mixture of lemonade and sour cream. The texture can also be offsetting to some people since it can be slightly gritty. It is worth keeping in mind that Airag is milk and a source of nutrients. After a day of riding it can actually be quite refreshing, once acquiring a taste for it.
The first thing you will be served every time you visit a ger will be milk tea, which is essentially a cup of boiled milk and water, sometimes with a couple pieces of tea leaf thrown in for good measure. You might want to build up your tolerance by drinking lots of milk in preparation for your stay because they don't drink much else, except perhaps boiled water if you specially request it during a longer stay. Also, most traditional nomadic foods such as dried yogurt and the like require acclimatization to milk as well. Cold drinks don't actually exist in the countryside unless you intend to drink straight out of a river, generally not recommended.
If you are in Mongolia especially in the country side try their National Home Made Vodka. It's usually made from distilled yogurt or milk. It doesn't have any weird taste. After you have your first shot of the vodka you won't feel anything, but few minutes later it will get to your head. Most people in Mongolia usually drink this for medical reasons. First you heat up the vodka then put in a little bit of special oil which is also made from milk. Careful don't overheat it, you might get blind. Mongolians call their national vodka nermel areehk "distilled vodka" or changa yum "tight stuff". There are lots of Russian type Vodkas sold all over the country. The best ones are Chinggis Khaan vodka, Soyombo and Golden Chinggis.
In Ulaanbaataar you can find most of Western beers, from Miller to Heineken. They sell Budweiser -- not American Bud but the Czech Budweiser. Local beer, such as Chingiss, Gem Grand, Borgio or Sengur is fine.