In the late 1700s and early 1800s the British took the island of Sri Lanka in fear of the French taking the island, due to Napoleon's conquest over the Netherlands. From this point until 1815 the British attempted to take the island from the local people, only succeeding after they took Kandy. The British used the island to grown rubber, tea, and coffee as a small number of British land owners controlled this industry and used Tamil immigrants (from India) to work the fields, giving the island a larger Tamil minority.
Throughout the 1800s and 1900s the relations between the Sinhalese and British improved, most notably among the Sinhalese's upper class. Despite this, there was a growing demand for independence among the people, leading to elected bodies on the island and eventually a proposed constitution in 1944. However at the time Britain was at war with Japan in World War II and Sri Lanka was a strong military base for the United Kingdom. The war though, was fought by many locals so it also created a stronger push for independence, just as Britain needed the island more than it had in the past.
Sri Lanka finally gained their independence in 1948. However the new government essentially excluded the Tamils from the vote as the Sinhalese were a minority in most tea, coffee, and rubber growing regions. This led to a strong backlash by the Tamils, but politics continued on this path and in 1956 Sinhala was the only legally recognized official language. This led to a growing Tamil insurgency as well as growing communist movements fighting for the rights of the poor, which included the Tamil people. One of these communist groups finally rose up in 1971 when they revolted, but were overtaken by the government.
In 1979 and into the 1980s the government made numerous changes, including giving the Tamils more rights, but not to the extent they demanded. This led to the rise of the Tamil Tigers, a group that sought greater rights and even independence from Sri Lanka and the ethnic Sinhalese. This led to numerous backlashes, but little government interference as the violence was primarily contained to the people fighting the people and many consider the events a civil war.
To make matter worst, many of the imprisoned communist movement leaders from 1971 were released from prison in the late 1980s and many ran for office (without success) or got involved in the violence. At this time they became a Sinhalese nationalist group that also excluded the Tamils, leading to both this movement as well as Tamil uprisings.
These violence movements continued until 2009 when the government declared that they had defeated the Tamil Tigers, many of whom had retreated to the region of Tamil Nadu in mainland India. This has substantially subdued violence, however the Tamils still have lesser rights and arguments continue between the groups.