The Hadean Eon began when the planet Earth first began to form, about 4.5 billion years ago. At first there was just a cloud of gas and dust, and then the Sun formed, and gradually the planets formed.
Around 45 million years after the planets first began to form, the Moon formed. Probably a large planetoid, about the size of Mars, crashed into the Earth. Little bits of hot rock splashed off during the crash and orbited around the Earth. Eventually these bits joined together, cooled off, and became the Moon.
On Earth, and probably on other planets too, the heavier iron atoms sank down and became the core of the Earth, and lighter atoms like silica and hydrogen rose to the surface. Most of the gases - hydrogen, carbon dioxide, methane, ammonia, and nitrogen - floated away into space. Water, brought to Earth by comets that crashed into the Earth, boiled into steam because the Earth was still very hot, and formed a steam atmosphere around the Earth.
Around 4.4 billion years ago, most of the planetoids had gotten smashed up into dust or had become part of a bigger planet. There weren't any more of them to smash into the Earth. Now that planetoids weren't always smashing into them, the Earth and the Moon formed rocky crusts of silica all over them. The oldest Earth rocks and Moon rocks we know about both date to this time. These are igneous rocks like granite and quartz.
As the Earth cooled down, about 4.3 billion years ago, the steam in the atmosphere also cooled down and fell as rain on the Earth, and that made the oceans. By 4.2 billion years ago, Earth had land and oceans just as it does today, and plate tectonics may have already been moving the land and oceans around. The oceans in some parts of the Earth may even have been frozen into ice, as the North and South Poles are today. Inside the oceans, amino acids from space began to join together into the first proteins - not yet life, but a step along the way. Probably the earliest RNA molecules also formed at this time.
By now almost a billion years had passed. By about 3.8 billion years ago, the continents were beginning to form on Earth, and this marks the change to the next eon, the Archaean.