How did the Mexican-American War division foreshadow the Civil War?

The division after the Mexican American War began with the introduction of the Wilmot Proviso which argued that slavery would not be permitted in the territory gained from the war. The proviso was added to an appropriations bill, and was similar to the language of the old Northwest Ordinances which had forbidden slavery in the old Northwest. Wilmot, a freshman Congressman from Pennsylvania made a long speech in support of the proviso during which he exclaimed:

 God forbid that we should be the means of planting this institution upon it."

The proviso came up for a vote twice, was passed in the House, but defeated in the Senate.  It had long been believed that the slavery issue had been settled by the Missouri Compromise; in fact President James K. Polk had proposed extending the Compromise line into the new territories. The Wilmot proviso only ignited the debate anew. In response to the proviso, John C. Calhoun offered the Calhoun Resolutions which argued that the new territories were the common possession of the several states, and the provisions of the 5th Amendment against taking property without just compensation applied; therefore slavery could not legally be prohibited. Although Calhoun's resolutions never came for a vote, his position was soon adopted by Southern representatives in the ensuing debates over slavery.

The situation was perhaps best described by Senator Thomas Hart Benton who said that Wilmot's proviso and Calhoun's resolutions were like a pair of shears. One alone could do nothing; but together they could sever the ties of the union. His words were prophetic.

The division over the Mexican-American War can be seen as a prelude to the Civil War because it pitted Southerners against Northerners over the issue of slavery.

Before the Mexican-American War, the US was evenly divided between slave states and free states.  As the war approached, it was seen by many in the North as an attempt to gain more territory for the slave states.  Because of this, many Northerners (like Abraham Lincoln, who was in Congress at that point) opposed the war.  Thus, the divisions over the Mexican-American War (like the Civil War itself) were North-South divisions over the issue of slavery.

How did the Mexican-American War contribute to the conditions leading to the American Civil War?

The previous posts were quite accurate.  The acquisition of more land helped to perpetuate the slavery question.  Prior to the war, the Missouri Compromise had somewhat tabled the issue in terms of establishing an equal balance of slave and free states.  Yet, the compromise failed to account for the dynamic nature of both the issue and the nation.  The issue was too intense with feelings too passionately experienced on both side to simply be "tabled" with a compromise.  The nation was too dynamic with industrial energy as well as expansionistic desires to remain intact.  With the war and its eventual acquisition of more land, the question as to what status those states would become ended up reigniting the passion behind the debate.  The proposal for popular sovereignty to decide the issue again ignited intense discussion and debate about the nature of slavery, leading into the Civil War.

When the Mexican-American War ended, the United States received the Mexican Cession which included all of California, Nevada and Utah, and parts of Arizona, New Mexico, Colorado, and Wyoming.  With this huge land acquisition, the issue of slavery in the new territories was raised.  This led directly to the Compromise of 1850.  This compromise deepened the division between the North and the South.  First, California would be admitted to the Union as a free state, which upset the South.  Two new territories would be established, Utah and New Mexico.  These territories would be open to slavery through popular sovereignty, that is, the people of the territories would decide the issue of slavery.  This upset the people in the North because so much territory could now have slavery. The slave trade was banned in the District of Columbia which upset the South because they saw this as a first step towards abolishing slavery.  There would be a new, strict, fugitive slave law which upset the North because now by law they had to assist in returning slaves to their owners. These tensions created by the Compromise of 1850, which was passed because of the results of the Mexican-American War, helped create the conditions for the Civil War.