What event triggered the fighting between the and the south?

The election of Abraham Lincoln to the office of the President.

When the results of the electors, in the electoral college, became known South Carolina immediately declared its secession from the union. South Carolina then demanded the withdrawal of union troops from within its borders. 6 other states followed South Carolina's lead. The US continued building Fort Sumter in Charleston Harbor and attempted to reinforce and resupply the fort and 3 other positions. The commander at Fort Sumter abandoned two positions in the harbor and on imminent danger of being overrun, surrendered Fort Moultrie, withdrawing all the troops to Fort Sumter. The garrison at the fort consisted of two artillery batteries, 85 troops total and 43 workmen. An attempt by the union to reinforce Sumter with 200 troops by sea resulted in Confederate shore batteries driving off the ships. On April 12th, 1861, confederate forces began the bombardment of Fort Sumter and the fort surrendered on April 14th. Lincoln immediately called on the states to raise 75000 troops to put down the insurrection and 4 more states seceded.