What were the underlying and immediate causes of the Civil War?

Interestingly, slavery wasn't the biggest issue in question at the start of the Civil War. The main conflict that sparked the war was the Confederate attack on Fort Sumter in 1861. After this attack, the Confederacy seceded from the Union and formed its own nation to engage in battle with the Union.

The underlying issues, however, were much more complex. Slavery was a hot-button issue, particularly in regard to the new territories being added—there was great debate over whether slavery should be allowed in new territories. However, also at the heart of the issue was the complex debate over state's rights. The Confederacy was dedicated to the rights of the states more so than the control of the federal government, and they clung to their state-given right to continue practicing slavery. They believed that, by legislating where slavery was legal and illegal, the federal government was taking power away from the states and was therefore overstepping its boundaries.

The immediate cause of the Civil War was the attack on Fort Sumter by Confederate batteries in April of 1861. This led Abraham Lincoln to issue a call for troops to crush the rebellion, which in turn caused Virginia, Tennessee, North Carolina, and Arkansas to secede from the Union. After this event, the Civil War was on. 

The underlying causes of the war were more complex, but can be traced to the divisive issue of slavery. Long a simmering issue, it became particularly toxic in the aftermath of the Mexican War, which raised the question of the expansion of slavery into the western territories. This political issue was increasing powered by the moral force of the abolitionist movement in the North. While most Northerners were not abolitionists, many were beginning to resent what they saw as the South's attempts to control the nation's politics. In the South, on the other hand, there was a fear that Northern sentiment was turning against them, and that if the federal government fell into the hands of antislavery men, they might move against slavery. The fact that the industrial North and the agricultural South were drifting apart culturally and economically exacerbated matters--slavery, in fact, was a leading cause of this divergence as well.

The Kansas-Nebraska Act, which permitted popular sovereignty (a vote) on the issue of slavery in Kansas, where slavery was previously prohibited, destroyed the so-called second two-party system, introducing a new party, the Republicans, devoted to resisting the expansion of the institution. When Abraham Lincoln, a Republican, was elected in 1860, South Carolina led the states of the lower South out of the Union, which set the stage for the more immediate causes. 

In a few words, the immediate cause was secession and firing on Fort Sumter, and the underlying cause was the political issue of slavery.

What were the short-term causes of the Civil War?

Short-term causes of the American Civil War certainly include Lincoln's refusal to meet with the commissioners that were sent by the Confederate government to buy Fort Sumter and other federal property that was located in the Confederacy.  This was followed by Lincoln's launching an invasion fleet from New York to invade South Carolina.  The Confederacy responded to this by subdueing Fort Sumter so that the invasion fleet could not land.

Upon the election of Lincoln to the Presidency, six southern states seceeded and formed the Confederate States of America.  In those days, secession was a Constitutional right, and  most citizens of America, both northern and southern, so believed.  When Lincoln called for the states of the union to send soldiers to conquor the states that had seceeded, seven more states seceeded.  (Two of these states had strong factions that remained loyal to the Union so that both sides claimed them.)

So we have it in order: Lincoln's refusal to sell federal property in the Confederacy, Lincoln's launching of an invasion fleet, and Lincoln's call for troops to suppress Constitutional rights.

But why did Lincoln direct events towards war?  If the Republican party and its president had earned the reputation of being the party and the president who presided over the dissolution of the Union, they would never have had a chance of being relected to power.

Much of this is discussed on pages 345-378 of vol. II of A Constitutional View of the War Between the States by Alexander H. Stephens (ca. 1870).  This book is still admired by American Constitutional scholars.

Not exactly sure what you mean by "short-term", but if I take it to mean the more immediate causes--the match to the gasoline as it were--then Confederate troops under Pierre Gustave Toutant Beauregard firing cannons at Fort Sumter in Charleston Harbor, South Carolina in April of 1861 would be the starting point.

In the months leading up to that battle, not only the election of Abraham Lincoln in November of 1860, but his election with a minority of the vote (40%) and without a single southern vote (his name wasn't on the ballot in ten states) confirmed Southerners worst fears: that the North could dictate to them through their larger population and representatives in Congress.

So they seceded.  Keep in mind that the war started in April, and Lincoln's election was in November, so Lincoln was pretty patient while he waited to take office, hoping a new compromise could be reached and crisis averted, but the number of states leaving the Union continued to grow, and with Fort Sumter, there was no turning back.

One of the major causes of the civil war had to do with the expansion of the United States and the conflict over whether new states would be admitted to the Union as slave states or free states. Though the Missouri Compromise was considered to be a partial solution, many people felt that it set in motion an unstoppable slide towards the war.

Nat Turner's slave rebellion in 1831 also caused Virginian lawmakers to remove some basic civil rights that had been accorded to slaves contributing to further unrest.

The Compromise of 1850, intended as a further band-aid to problems inherent in the westward expansion of the United States also pushed the country closer to open conflict.

John Brown's raid in 1859 coupled with the election of Abraham Lincoln and his Republican Party's anti-slavery position led to the secession of South Carolina which was then followed by other southern states in 1860.

An attempt to re-supply Fort Sumter triggered a Southern response, the first open battle of the Civil War that would lead to the surrender of Fort Sumter and the onset of the Civil War.

The American Civil War had both long term and short term causes.

In the long term, the war was caused by major differences between the North and South.  The biggest difference was, of course, slavery.  But the two sections were also very different in terms of their economic systems and their cultures.  These differences caused continual friction between the two regions.

In the short term, we can look at two relatively immediate triggers for the war.  The first was the election of Abraham Lincoln.  Southerners thought he would be an enemy to them and secession followed soon after his election.  The second was Lincoln’s decision to resupply Fort Sumter and the South’s decision to attack the fort.

What were the primary causes of the Civil War?

While it is true that the war was not specifically fought over slavery, I think that it is going too far to say that slavery was “just a small part” of what caused the war.  Slavery was at the heart of the causes of the war.

The major cause of the war was that the North and the South did not trust one another.  Each thought the other side was working to destroy the other’s way of life.  While there were many aspects to these different ways of life, the most important difference was slavery.

The North and the South had very different social and economic systems.  The South was based on a plantation economy that raised staple crops.  Of course, the work force for this economy was enslaved.  The North had a mixed economy based on free labor.  The South was an aristocratic society headed by plantation owners.  The North was much more egalitarian.  In these ways, the two were sort of like separate countries that were officially united. 

Because the two “countries” were so different, they were each suspicious of the other.  They each tended to think that the other was gaining the upper hand and would destroy their way of life.  The war was fought mainly because each side wanted to preserve its way of life, but the differences in their ways of live were largely centered on slavery.