One must understand that the so-called Eelam wars were not mere spontaneous outburst of gunfire from a minority community. I'll attempt to briefly outline the series of events that led to this conflict of tragic proportions. This is by no means exhaustive nor do I claim so. This is my understanding of the chronology of the events as they unfolded
Historically Tamils and Sinhalese lorded over independent domains in pre-colonial Ceylon, which were unified under the British rule for administrative convenience.
The roots of the conflict lie in systematic attempts to marginalise the Sri Lankan Tamils since Sri Lanka (then Ceylon) gained independence from the British in 1948. Prior to independence, Ceylon Tamils or Illankai Tamils, who were native inhabitants of the island (not to be confused with Upcountry or plantation Tamils who were chaperoned from India to work in British-owned tea estates and plantations in the Central highlands) dominated the civil service and the upper echelons of the bureaucracy, owing to their better education and understanding of English.
Post-independence Solomon Bandaranaike in 1956 imposed Sinhala as the sole official language in place of English, which was naturally protested by Tamils, who insisted (largely peacefully) on a federal setup and parity for Tamil in official matters. The year long protests lead to the signing of Bandaranaike-Chelvanayagam pact, an agreement reached between Ceylon govt headed by Bandaranaike and Tamils represented by S.J. Chelvanayagam, which promised to address Tamil concerns about language and autonomy and nip the communal flareups in the bud. The pact was however, abandoned by Bandaranaike in 1958, who himself was assassinated a year later.
Following the 1960 elections, Bandaranaike's widow Sirimavo Bandaranaike came to power on her own. Consequently, Sinhala was proclaimed as the language and Tamils were discriminated in most walks of life, culminating in the the Vaddukoddai Resolution of 1976 by Tamil United Liberation Front (TULF). TULF contested the 1977 elections on the plank of Tamil sovereignty and Eelam for Illankai Tamils. Following this SL govt. passed an amendment requiring MPs to take oath solemnly affirming their commitment to a united Sri Lanka. TULF MPs refused to do so and were thrown out of the parliament. This spawned a host of other Tamil separatist groups such as People's Liberation Organisation of Tamil Eelam (PLOTE), Tamil Eelam Liberation Organization (TELO), Eelam People's Revolutionary Liberation Front (EPRLF), Eelam Revolutionary Organisation of Students (EROS) and Liberation Tigers of Tamil Eelam (LTTE) who took to militancy to achieve their goals. These groups fought with SL government and with each other for supremacy. Political assasinations were the order of the day. In 1983, LTTE ambushed and killed 13 SLA soldiers in the North, this was followed by Black July pogroms in Colombo where Tamils and their assets were targeted, burnt, looted and decimated with active support of the SL govt.
Fearing repercussions in Tamil Nadu and the then SL governments pro-Western stance, India began arming and training Tamil militants to gain some measure of the situation. Meanwhile LTTE went from strength to strength as they massacred their rivals and civilians (146 civilians were massacred by LTTE in Anuradhapura in 1985) and SL govt too actively contributed to the retaliatory killings of Tamils. Following the failure of peace talks between SL govt and LTTE, SL government launched a military campaign to suppress the Tamil groups and laid siege to vast areas in the North. India in response airdropped food parcels and medical supplies over the sieged areas particularly Jaffna.
The Rajiv-Jayawardene accord was signed in 1987, which aimed to bring peace by making concessions to the Tamil minority- merger of the North-East, inclusion for Tamil as the second official language, promises of devolution of powers to the new province etc. India in turn agreed to send a peace keeping force (IPKF) to establish order in the war zone, cease assisting the separatist groups and disarm them. Naturally this invoked opposition from both Sinhalese and Tamils and they came together in a sort of unholy alliance against the IPKF. Not to put too fine a point on it, IPKF mission was a disaster and we lost 1200 soldiers with little to show for it, not to mention a less than graceful exit in 1989.
Following this LTTE moved to occupy areas vacated by IPKF and established their unchallenged dominance across much of the Tamil-populated areas. The brutalities they inflicted in the process were horrific- the biggest being the massacre of nearly 800 surrendered policemen in the Eastern province. Rajiv Gandhi's assassination in 1991 brought an end to all covert support for LTTE in India and India washed their hands of the matter.
With the ascent of Chandrika Kumaratunge to power in 1994 initiated a new series of military campaigns against LTTE (broadly recognised as Eelam war III) and succeeded in driving them out of Jaffna into Vanni region. However, LTTE responded with Oyatha Alaigal (unceasing waves) and the two sides frequently exchanged territories including key points such as Kilinochi, Mullaitivu and Oddusudan and Mathu, before LTTE decisively gained control over much of the North, especially Elephant Pass Garrison the key link between Jaffna and the mainland. However, they were unable to capture Jaffna itself.
In 2002, a new peace process was initiated with international interlocutors aimed at finding a peaceful solution to the conflict. Inspite or promising beginnings, the peace process floundered and was often violated by both sides. The split in LTTE in 2004 weakened them in East and encouraged the government to give the peace process a slip. A series of armed clashes and failed political maneuvers led to Eelam war IV, which saw unprecedented blood shed and atrocities by both sides. Slowly but surely, SLA recaptured LTTE held areas in the East and the North and boxed the LTTE combatants into ever shrinking area in the North. Tragically Tamil civilians forced as human shields by an increasingly desperate LTTE leadership were squeezed from both sides and faced innumerable sufferings.
The Eelam war finally ended on May 18th 2008 when Prabhakaran was shot dead. A war where as per conservative estimates over 100,000 killed were citizens of the same country. A pyrrhic victory which brought along unfathomable pain and suffering to the citizens of Sri Lanka- Tamil and Sinhalese.