The most obvious colonial problem that a viewer can perceive from viewing Lagaan is the fact that the British Empire imposed taxes on rural individual villages during the colonization of India.
The village in which the story takes place has experienced poor weather for farming. This makes the villagers unable to pay their taxes to the regional leader of the British Raj, Captain Russel.
In the film, this problem is solved when the protagonist, Bhuvan, accepts a challenge from Captain Russel to play a cricket match. The villagers, despite being underdogs, win the match and, as a result, do not have to pay their taxes, which brings them economic security and a sense of freedom. (The word lagaan is Hindi for "taxes.")
Another interesting colonial problem that comes up in the film is the British Raj's technique of "divide and conquer" as a means for colonialism. India is a country with a large number of ethnic groups. The British were able to create tensions between these ethnic groups in order to make the colonization process easier for them.
An Indian man named Ram Singh serves as an assistant to Captain Russel. Though he is an Indian, he has been separated from his countrymen to work for the British. In a dramatic moment late in the film, Ram Singh rejects Captain Russel and marches toward the Indian villagers, symbolizing his loyalty to his homeland.
Another example of the "divide and conquer" strategy that comes up in the film is the character Ismail, who lives in the village with Bhuvan and the other aspiring cricketers. Ismail, a Muslim, is at first reluctant to join the Hindu villagers on the cricket team. Historically, the conflict between Muslims and Hindus was a lot more serious than what is depicted in this film, but the characters in the film overcome this problem to win the match and their economic freedom.