The other answers here do a good job of explaining how the alliance system made a major conflict all but avoidable. One detail that made the alliance system between the European powers particularly dangerous was that some of it was conducted in secret. In particular, Germany and the Austro-Hungarian Empire had a secret treaty in place that assured Germany would come to the aid of its southern neighbor in case of an attack by Russia and vice-versa. This arrangement had been made in 1879 but was not tested until Russia attacked the Austro-Hungarians in 1914.
While the existence of this alliance was leaked in 1883, the details of it remained secret. This might explain why Russia did not consider the Germans to be much of a threat when they began their attacks on the Austro-Hungarians.
In retrospect, keeping the details of such military alliances a secret was a bad idea. Alliances of these kinds are best utilized when they are meant to prevent aggression from non-treaty members. We will never know for sure, but it is possible that the Russians would not have been so quick to attack Austria-Hungary if they knew what the full extent of Germany's response would be.
All in all, the many military alliances between the various European powers all but assured a major war. It made it nearly impossible for WWI to have been prevented once fighting initially broke out. Considering that each alliance had little idea of exactly how the alliances of their potential foes functioned, they ended up rushing into a situation that would consume the entire continent in war.
The Alliance system was a key factor in the causation of WWI. Germany was on alert because it was surrounded by two opposing allied powers, France and Russia. Russia allied itself with Serbia because Czar Nicholas II viewed himself as the protector of the Eastern Orthodox faith and all of the Slavs. Britain joined forces with France and Russia because it feared a strong Germany, especially German naval improvements. Germany allied itself with the Ottoman Empire in order to potentially threaten British interests in India. Germany also allied itself with the Austro-Hungarian Empire because that empire had many German-speaking people.
These alliances, while somewhat complex, made what should have been a regional war explode into a war that would engulf Europe and the world. A Serbian nationalist, whose ties to the government are still debated to this day, shot the Austrian archduke Franz Ferdinand. Austria-Hungary issued an ultimatum to Serbia which it knew it would not accept. Serbia looked to Russia for protection, and Russia soon started to mobilize. Germany, fearing an attack from Russia, declared war on Russia—this pulled France into the war. Britain was not obligated to help France or Russia fight its wars, but when Belgium was invaded by Germany, Britain issued an ultimatum to Germany to leave the tiny country. Germany refused and this pulled Britain into the war. The Ottoman Empire declared war on Russia (and thus the rest of the alliance) in an attempt to regain land lost to Russia in the Russo-Turkish War of 1878.
Before World War I, many of the major European powers were involved in two strategic alliances. The first of these two was the "Triple Alliance" or "Triplice." Formed in May of 1882, it consisted of Germany, Austria-Hungary, and Italy. The second opposing alliance was the Triple Entente, originating in bilateral treaties between France and Britain (the "Entente cordiale") and Britain and Russia.
In World War I, these alliances became the cores of the two opposing sides, with the Triple Alliance becoming the Central Powers and the Triple Entente becoming the Allied Powers.
The reason that the alliance system contributed to the First World War was that it meant that local conflicts, even minor border disputes, rather than remaining local in nature would draw in all members of both alliances. Thus conflicts between Austria-Hungary over Serbia drew in not only Russia, which had territorial ambitions in the area, but also their western European allies.
The Alliance system was one of the causes of World War I. Prior to the war there were two alliances. One was the Triple Alliance. This included Austria-Hungary, Germany, and Italy. The other alliance was the Triple Entente. This included Soviet Union (Russia), Great Britain, and France. The danger of these alliances is that if a member from one alliance declared war on a member from the other alliance, the conflict would quickly escalate. That is what happened in World War I.
When the next king of Austria-Hungary, Franz Ferdinand, was assassinated by a group of Serbian nationals, Austria-Hungary made a series of demands on the government of Serbia. When the government couldn’t meet all of these demands, Austria-Hungary, with the support and approval of Germany, declared war on Serbia. Since Serbia and Russia were very close friends and allies, Russia declared war on Austria-Hungary. Germany then declared war on Russia. France and Germany then declared war on each other. Eventually, Britain declared war on Germany when Germany invaded the neutral country of Belgium. Thus, what was originally a conflict between Austria-Hungary and Serbia quickly escalated into World War I, involving many countries throughout the world.