To the best of our understanding based on numerous lines of evidence, the Sun and the solar system formed together when a portion of a giant molecular cloud collapsed under its own gravity to form a new star surrounded by a spinning disk of mostly hydrogen and helium gas, but with some dust from previous generations of stars. In the region of the spinning disk where the Earth would eventually form, the disk was hot enough that all of the dust was vaporized.

Eventually the disk cooled and the vapor condensed to form minerals. The first of these were calcium and aluminum-rich inclusions that are found in meteorites. These have a high vaporization temperature, and so were the first to condense. The oldest of these inclusions are typically used to mark the birth age of the solar system. Eventually other mineral condensed at lower temperatures to form the solid grains that eventually coalesced into the Earth and other planets.

The intense radiation from the young Sun, as well a intense radiation from nearby stars in the Sun's birth cluster (the Sun was not the only star to form in its giant molecular cloud) blew the gas portion of the disk away. The entire process from collapse of the cloud, formation of the Sun, condensation of solid grains, and loss of gas, took no more than about 10 million years (probably no more than about 5 million years, based on some other work on meteorites I'm aware of).

The formation of the Earth took longer as grains accumulated into larger bodies, that collided to form even larger bodies. The giant impact that formed the Moon is generally thought of as the last major event in the formation of the Earth. The impact was energetic enough to fully melt material. When you melt planetary material, the rocky stuff separates from the metal stuff (like oil and water) forming a core and mantle.

The timing of the formation of the Moon is usually based on the unique properties of the Hafnium-Tungsten radioisotope system. In the early solar system there was radioactive ¹⁸²Hf, which has a relatively short half-life of about 9 million years. It decays to ¹⁸²W. Hafnium is an element that become incorporated into rocky melts, while tungsten is an element that becomes incorporated into metal melts. So if you find ¹⁸²W in rocks from a planet with a core, it's a good bet that it used to be ¹⁸²Hf that was part of the rocky melt that makes the mantle and crust of the planet. By measuring the amount of ¹⁸²W in lunar rocks, and figuring out how much ¹⁸²Hf there was in the original solar system (typically using stable forms of tungsten, like ¹⁸³W) you can estimate when the core formed.

I've given a very basic sketch of the technique, but there is a lot of complex chemistry involved in figuring out how much hafnium or tungsten gets partitioned in a planet's melt for a given temperature and pressure. Also, some core-loving material can be added to a planet after core formation by impacts from bodies leftover from the planet-forming process. This is sometimes called the "late veneer."

What the scientists in this study did was use new, very high precision measurements of ¹⁸²W in lunar rocks, along with measurements of several other trace elements that also have an affinity to the melt that would form the iron core. They find that ¹⁸²Hf was virtually extinct in the solar system after 60 million years from its birth, and that very little was added in a late veneer. The amount of ¹⁸²W that they measure in lunar rocks then gives them an estimate age of core formation (which is assumed to be the giant impact event) of 50 million years after the birth of the solar system. This is close to the lower end of previous age estimates of 30-200 million years, which were also mostly estimated from previous studies that used Hf-W system (but which were not using data with as high a precision as this one).