It was a cataclysmic conflict which is deliberately left vague, but was evidently fought between the American Empire and Greater China. It is likely that the Middle East was involved, as this region has been reduced to a flooded maze of atomic craters.
The name suggests that the war took only an hour to begin and end, presumably due to the unprecedented amounts of devastating weaponry deployed. The war ended millennia of human civilisational development, and caused near-total global destruction, depopulation, realignment of the continents, massive geological upheaval, and destruction of several large landmasses, and the downfall of the human species. However, it is likely (though unconfirmed) that there was a cold war leading up to this event and that the "sixty minutes" only refers to the actual exchange of WMDs.
The scientific and technical knowledge of most of humanity's advanced technology was lost during the conflict, and the world collapsed into a post apocalyptic state with the level of technology still not recovered to pre-war status centuries or millennia later.
Following the war came a period of immense geological upheaval, as earthquakes, tsunamis, glaciers and volcanic activity swept across the world; the isthmus linking North and South America was destroyed by weaponised meteors, volcanoes sprouted up in Europe and new oceans and mountain ranges were formed. It is uncertain whether this was a coincidence or unintended side-effect due to the immense energies released during the war, or if it was intentionally caused by some of the weapons employed.
North America was reduced to a radioactive wasteland, along with much of eastern China. The surviving Chinese fled into the Himalayas and formed a new civilization that would later become Shan Guo, leading nation of the Anti-Traction League. Africa escaped much of the devastation and parts of the continent became the center of another new civilisation though it was weakened by the Traction Cities centuries later.
This destruction led to the so called "Black Centuries", an era presumably similar to the historical Dark Ages. During this period, barbaric nomad tribes wandered Europe. After this period (at least 10,000 years later), Nicholas Quirke transformed London into the first of the Traction Cities and this led to the rise of Municipal Darwinism and the Traction Era.
Remnants of weaponry
Particle compressing weapons, quantum energy weapons, biological weapons and conventional nuclear weapons, some deployed upon orbital platforms, were utilized to great extent. More esoteric weapons may also have been used or at least developed; the American Empire is mentioned in passing to have possessed weaponry capable of somehow drawing power from parallel universes, though this is an in-universe rumour from a world many years after the War itself destroyed civilization so it might not be accurate. Whatever was actually employed has survived in legend as being powerful enough to have "made the gods quake in fear" and was capable of projecting at the very least megaton-level firepower given that the few weapons actually seen in action in the books regularly destroy entire cities with single shots.
Several items featured in the series are deadly relics of the conflict, including a directed-energy weapon called MEDUSA (Military Energy Department, United States of America) which is later mounted upon the Traction City of London. This weapon (and the computer that controls it) was recovered by a group of explorers from an ancient military base on the American continent. MEDUSA is a key element of the plot in Mortal Engines.
Another featured weapon was the device known as ODIN (Orbital Defence Initiative), another ancient American weapon consisting of a particle cannon mounted on a satellite in Earth's orbit. ODIN is a key element of the plot in Infernal Devices and A Darkling Plain, as several major organizations (particularly the Green Storm) attempt to gain control of the device to destroy their enemies. Other Ancient orbital weapons mentioned in passing are the Diamond Bat, Jinju 14, and the Nine Sisters, although these take no part in the plot and are not described beyond their name. It is implied that they are likely to have fallen out of orbit or to have ceased functioning during the long millennia between the Sixty Minute War and the rise of the Traction Cities.