The betel is type of vine which is native to South and South-East Asia and which is commonly chewed to produce this juice. Orwell makes reference to betel juice in the first paragraph of "Shooting An Elephant" when he is discussing the treatment of white colonials by the native Burmese population. Specifically, Orwell describes how the Burmese would spit this juice all over the dress of a white colonial lady as she walked through the bazaar.
This is significant because the act of spitting on another person is direct evidence of a deep-seated distrust and contempt. By spitting on a lady, the Burmese vented their dislike of the colonial system and, by doing it in the confines of the bazaar, were less likely to be caught and punished by the colonial police. This act, then, should be viewed in the context of Orwell's earlier comment that the Burmese did not openly rebel against their colonial masters ("No one had the guts to raise a riot"): they acted in a way which made their contempt clear but would not arouse the suspicions of the colonial police.
Betel juice, also called betel nut juice, is the juice that is produced when a person chews betel nut. Many people in Asia (mainly Southeast Asia) chew betel nut, as do many people in parts of Oceania like Palau and the Federated States of Micronesia. Betel nut is typically wrapped in betel leaf and powdered with slaked lime (gotten from coral) and then chewed. As the user chews, a juice is produced that must be spit out constantly. The juice is red, though Orwell describes it as “brown” in “Shooting an Elephant.” Here is a picture of a betel nut juice stain.
In the story, betel juice is mentioned only once, at the very beginning. It is used to show that the people of Burma resented the fact that their country was controlled by foreign imperialists. In the first paragraph, he says that
…if a European woman went through the bazaars alone somebody would probably spit betel juice over her dress.
Betel juice would stain the dress. In other words, the Burmese would spit their betel juice on a white woman’s dress to ruin it and to show their anger towards the white imperialists.