There have been numerous conflicts between the colonies throughout history, but these are the major ones that influence current affairs.
This is not meant to be an all-inclusive list, but please check with staff before inventing major conflicts between the colonies.
Virgon and Leonis were the two great empires, controlling 8 of the 12 colonies at the height of their influence. (Only Caprica, Gemenon, Picon and Aquaria were never conquered by either empire.) The colonies were at war for centuries, with various planets changing hands or becoming divided between them at various points.
The constant strain of war and colonialism ultimately led to the collapse of both empires. Over the past 200 years or so, all of their former holdings (except Sagittaron) have gained independence. Virgon and Leonis are officially at peace now, but there are still hard feelings between these long-standing enemies.
The conflict between Caprica and Tauron started during the Tauron civil war, about 100 years ago. There were allegations that Caprica was secretly funding one of the factions in the war (the one more advantageous to Caprica, naturally).
Though the accusations were never proved, the (allegedly) Caprican-backed faction lost the war and the winning faction never forgot - or forgave. Ever since, the colonies have been in an escalating cold war. There have been various issues that threatened to ignite the conflict into a full-blown war - a border dispute over an asteroid, claims of espionage, and posturing on Sagittaron, etc. - but so far no overt actions.
Sagittaron has been fiercely contested for centuries. It is now a hotbed of civil unrest and political/economic wrangling by the other colonies.
Canceron and Scorpia have both gained their independence within the past 60 years. They do not look favorably upon their former colonial oppressors, and also suffer from the internal strife that comes from trying to establish oneself as a new nation.