Cows do contribute to global warming, although in fact they mostly do so by burping rather than farting. Cows are ruminants--they specialize in eating plants like grasses that are difficult to digest, so they have evolved a four-chambered stomach--they chew their food, begin digesting it in their first two stomach chambers, then regurgitate this "cud" and chew it again,before passing it back to their stomach and then intestine to complete digestion. Like a lot of other animals (including humans!), ruminant shave a complex association of microbes living in their digestive system to help them break down their food. One of the microbes that lives in their stomachs makes methane (CH₄)--the same gas that we burn as "natural gas." Methane is a strong greenhouse gas--it's about twenty times stronger than carbon dioxide, although over a few years methane in the atmosphere will react with oxygen and become transformed into carbon dioxide. In the US, about 20% of human-related methane is due to cows and other ruminants, so livestock do contribute to global warming. People are now trying to develop different cattle feeds that result in less methane production.

It's actually the cow "burps" that mostly contribute to global warming more, although the "farts" cause problems as well.Because of how cows digest their food, they release a lot of methane through belching (and a smaller amount through farting), and nitrous oxide and ammonia in their manure. Like carbon dioxide, methane is a greenhouse gas, but methane traps heat in the atmosphere even more effectively than carbon dioxide does. It's become a problem because we have so many cows and other livestock to feed us. (All the ammonia and nitrous oxide cows make can cause big ecological problems too.) A lot of people think everyone needs to cut back on the amount of meat we consume to help reduce the greenhouse gases made by livestock animals. Also, if the same amount of space were used to grow crops instead of growing livestock, it's thought that we would be able to feed more people, since growing crops utilizes space and resources better than growing livestock.

For more information on livestock and their effect on greenhouse gases, see these websites:

Technically, no - the methane is actually burped, not farted. Methane is a greenhouse gas, in fact a quite powerful one, but still a very minor one compared to the two more common major greenhouse gasses, carbon dioxide and (most important of all) water vapor. From what I understand, we still aren't entirely certain just what the role of carbon dioxide is in global warming, so I think it's safe to say that we can't really say what the significance of methane is either.