This self-referential colloquialism was created in the United Kingdom during the early 1960s to late 1980s and, in the UK is sometimes pronounced "Dezi."
In this sense, it can include:
Within India, desi connotes "local" or "indigenous", as distinct from videshi or pardesi ("foreigner").
During the heyday of the British Raj, many people from the then undivided Indian sub-continent emigrated to the UK or to other British colonies, in search of education and opportunity. The diaspora from what is now called South Asia increased dramatically following the riots and massacres of Partition. Families from the affected areas sought safety in various Commonwealth countries. Starting in the 1960s, the USA dramatically increased the amount of immigration permitted from Asia, leading to large immigration from the subcontinent.
Likewise with the split of the East Pakistan Wing from West Pakistan in 1971 needed a name by the Awami League and colloquially the Bengalis (both East and West Bengal, even before partition) had referred to their homeland as 'Bangla-desh' meaning the homeland of Bengalis. The National Assembly then adopted 'Bangla Desh' as the name for the country.
Communities that have remained distinct in South Asia have tended to mix in diaspora. Some second or third generation immigrants, but not all, do not think of themselves as belonging to a particular nation, sub-culture, or caste, but as just plain South Asians or Desis. Some Desis are creating what can be called a "fusion" culture, in which foods, fashions, music, and the like from many areas of South Asia are "fused" with elements from Western culture.
It may also mean "native" or "traditional." Common examples are "desi ghee," which is the traditional clarified butter of the Asian subcontinent, as opposed to more processed fats such as vegetable oils. "Desi chicken" may mean a native breed of chicken. Heritage varieties of vegetables and other produce can also be qualified as "desi."
The dish chicken tikka masala is also said to be invented by a Bangladeshi chef in East London after a customer asked for gravy with his chicken tikka.