We have seen many wars in Star Trek, but which were the major ones? For the purposes of this question, consider "major" to be something similar to WWI or WWII — conlicts between superpowers that take a lot of time and is not limited to a few skirmishes.
As a time range, take the time between the founding of the Federation until Star Trek: Nemesis.
Some of the major wars in the galaxy between the founding of the Federation and Nemesis were:
Federation - Klingon Empire Conflict (late 2100s to 2293): This century-long conflict featured periods of both open warfare and implicit hostility between the two powers, including the Federation-Klingon War of 2267. A Neutral Zone was established that was eventually dismantled when hostilities ended formally, as part of the first Khitomer Accord of 2293. Peace between the two parties reshaped the balance of power in the Alpha Quadrant. Elements of this conflict are chronicled in The Original Series, and in the films The Search for Spock, The Voyage Home, The Final Frontier, and The Undiscovered Country. An important event was the Battle of Caleb IV, a decisive victory for the Klingons, as recounted in the Deep Space Nine episode "Once More Unto the Breach".
Klingon Empire - Romulan Star Empire Conflict (late 2200s to mid-2300s): A series of violent incidents between these two empires, which included the Khitomer Massacre that left Worf orphaned. This is mainly discussed in The Next Generation, in the context of the Klingon Civil War and Worf's restoration of honour.
Federation - Cardassian Union War (2340s to 2367): A lengthy two-decade war that saw much bloodshed, including the Massacre of Setlik III in 2347. (Miles O'Brien served in the Battle of Setlik III, seventeen years before his posting to the Enterprise, and was remembered as the "Hero of Setlik III".) The war would have lasting political repercussions for the Alpha Quadrant, including the uprising by the Maquis and fuel for the Dominion War. This war is discussed in The Next Generation and Deep Space Nine.
Klingon Civil War (2368): A conflict between Gowron and the Duras family for control of the Empire, in which Worf and his brother played a role. This is chronicled in The Next Generation.
Klingon Empire - Cardassian Union War and Federation - Klingon Empire War (2372 to 2373): A sequence of brief but bloody wars, partly caused by a Changeling from the Dominion infiltrating the Klingon military hierachy. These are documented by Deep Space Nine and lead into the Dominion War.
Federation - Dominion War (2373 to 2375): A relatively short but very costly war, that almost saw the Federation's defeat. By its end, it also involved the Klingon Empire and the Romulan Star Empire on the side of the Federation, which helped to turn the conflict in the Federation's favour. This war is seen mostly in Deep Space Nine, which chronicles the events leading up to the war and the war itself. References to it are made in Insurrection and Nemesis (the latter of which occurred just a few years after the war).
Of course, there were other conflicts, but many were minor in comparison to these or difficult to label as a "war" (e.g. the Federation-Borg conflict, which was mainly a series of incursions by the latter).
A complete list of all conflicts in the Star Trek universe can be found here.
Note: While the United Earth - Romulan Star Empire War was a significant conflict that had galactic repercussions (in particular, the formation of the Neutral Zone between the Federation and the Romulan Empire and a lasting distrust between the two powers), this war ended in 2160, a year before the founding of the Federation.
