Austria-Hungary started the war when they used the assassination of Archduke Franz Ferdinand by a Bosnian Serb nationalist as a pretext to declare war on Serbia. After the assassination, Austria sent an ultimatum to Serbia on July 23, 1914. This ultimatum was only sent after the Austrians received confirmation from Germany that they would support them in the crisis. You can read the ultimatum here, but the essence of the demands was that Austria-Hungary wanted considerable influence in Serbian domestic affairs. Serbia's response was to agree to some of the demands and ask for clarification and negotiation on others, and only one demand (that Austro-Hungarian officials participate in legal proceedings against Serbian nationals on Serbian soil who were allegedly involved in the assassination) was outright refused. On July 28, Austria-Hungary declared war on Serbia. Russia, a staunch ally of Serbia, fully mobilized its army in preparation for war on July 30. Between August 1 and August 4, Germany declared war on Luxembourg, Russia, France, and Belgium. France was an ally of Russia and Germany intended to invade France through Luxembourg and Belgium. Britain, which guaranteed Belgium against foreign invasion, declared war on Germany on August 4.

The Austro-Hungarians never intended for Serbia to accept the ultimatum, and pro-war Austro-Hungarian leaders believed that a war with Serbia would expand their influence in the Balkans, improve their international prestige, and reduce ethnic tensions in the empire (on that last point, it was widely believed that Serbia was to blame for unrest among Southern Slavs living in the Empire, and that eliminating Serbia would remove the source of these tensions).

Germany is often blamed for instigating the war because they offered Austria-Hungary support for invading Serbia (the "blank check" you hear so much about) and because they expanded the conflict by preemptively declaring war on France and Russia. It is worth noting that Germany was by far the strongest of the Central Powers, and that the country's eventual leadership of the alliance played no small part in the postwar narrative that Germany was to blame.

Sticking to immediate causality, World War I began when Austria-Hungary declared war on Serbia. Whether you blame Germany, or any other country for starting the war has a lot to do with your own views and interpretation of these events - this is an incredibly contentious subject and there have been many different arguments made over which countries, governments, and individuals are most to blame for the outbreak of the war. I will conclude by saying that the decisions which led to war were mostly made by unelected leaders, that the decision-making apparatuses in these countries were flawed and often resulted in miscommunication, and that the belief that a major European war would benefit the nation was not limited to those in Austria-Hungary or Germany.