I haven't heard these terms much in America, but it is the way that Indians refer to being vegetarian or not vegetarian. I thought I'd talk a bit about the history of vegetarianism in India and Hinduism today.
Personally I been feeling very drawn to being Veg, my body seems to want it. For the last week or so I have stopped eating meat except for fish. For the time being I am still eating fish and animal products like eggs and milk. So I can't really call myself Veg yet, but I'm closer anyway.
According to Wikipedia, India is 31% vegetarian (this being something called lacto-vegetarian, which includes eating diary products except for eggs). Another 9% eat eggs. That number seemed low to me. I was surprised. Although Wikipedia seems to say that only 4% of Americans identify as vegetarian, so that puts the number in perspective. Then again, the article says some estimates put the number of Indian vegetarians at 20-42%. I wonder if the variation has to do with the definitions being used for vegetarian. In America, I know, there is a lot of variety of which things can be eaten.
I think in America we are often less cautious about animal products that wander into unexpected places. Gelatin in a variety of foods, chicken broth used for vegetable soups, or MacDonald's fries that have been fried in animal lard. It depends on the individual how strict she is going to be. According to the article, Indians are much, much more cautious about what is in food and packages of food and medicine are marked according to whether they have any animal product in them.
Two main reasons given for vegetarianism in India are 1) Ahimsa and 2) the belief that what we eat effects our personalities, minds, souls ("You are what you eat").
Ahimsa is a Sanskirt word meaning non-violence. It is listed by Krishna as one of the qualities of a great man. I've heard several people say that ahmisa is much more stressed in Buddhism than in Hinduism, but Hinduism has started to be affected by it through Buddhism. A history of ahimsa can be found here.
The wikipedia article on vegetarianism in India says: "Also, Hindus believe that one's personality is affected by the kind of food one consumes and eating flesh is considered bad for once's spiritual/mental well-being. It takes many more vegetables or plants to produce an equal amount of meat  many more lives are destroyed and more suffering is caused when meat is used as food ."
The analysis of the statistics say that the percentages of vegetarianism are much higher for Jains and Brahmans (55%).
Jainism is an off-shoot of Hinduism. Sometimes it is considered a separate religion and sometimes it is considered a branch. The standards of ahimsa are extremely strict. Ideally only fruits and vegetables that have naturally fallen to the ground can be eaten so as not to harm the plants by pulling things off.
Some commenters have mentioned before that different degrees of Veg or non-Veg are expected for different castes. This seems to relate to the idea that what we eat is related to who we are and what we do.
Hinduism is a religion that is beautifully grounded in consequences. A good reason for Veg diet and ahimsa is the law of karma (law of action). It is better for our sanskara to limit the amount of killing or suffering we are causing.
Also, getting a bit off topic, the belief in reincarnation makes the consequences of everything more important. The long-term sustainability of a meat-less diet and the help to the environment is really important because Hindus cannot just shrug and figure that the long-term consequences won't effect them. Because we live over and over again, the damages that happen to the environment now are things we will have to deal with ourselves when we are reborn, not just something our children or grandchildren will have to deal with.
Now, in Vedic times animal sacrifices were part of the larger worships. The way that animals were killed is called jhaṭkā . The article doesn't say, but presumably this is a method of killing an animal which is more humane. One of the concerns with eating meat is that the fear that the animal felt in its last moments of life would be transferred into those who ate it. Meat that is sometimes eaten is often lamb meat. Never beef for any Hindus, as far as I know.
I'm sure that you've heard that the cow is sacred. Well, all animals and all people and all everything is sacred! The cow is special because the cow is so generous and provides us with the gift of milk, ghee, cheese, yoghurt, etc.
The transition from the culture of animal sacrifices to valuing ahimsa seems to give support to the idea that Hinduism was influenced by its off-shoot, Buddhism. Now, none of these articles has said what is common for those of the Kshatriya(warrior) caste to eat and I would suspect they were the ones doing most of the animal sacrifices in ancient times. And certainly there is plenty of hunting going on by this caste in the great epics.
Some Vaishnavas (one of the major branches of Hinduism) also stay away from garlic, onions, and other overly-stimulating foods. Someone mentioned here earlier that ISKON avoids those things and they are a Vaishnava organization. This has to do with the guna of food. I'll have to do a post on the gunas soon!
So, there is my overview of Veg and non-Veg. Please, feel free to comment and correct or clarify what I have found.