The "quick" version - made short by omitting large amounts of awesomeness, not by weeding out unimportant stuff. It's all important.
With the help of Tulkas, the First War was ended and Melkor had left Arda. There was peace as the Valar laboured, building up and bringing to order the lands and the seas, and causing green things to grow. Thus began the Spring of Arda.
The Valar made two lamps and they were set upon high pillars, Illuin in the north and Ormal in the south. Where the two lights met and blended, the Valar set up their first dwelling upon the Isle of Almaren.
The Valar rested and had a great feast, and because of light they had made they did not see the shadow of Melkor as he arrived in the north with his host.
Melkor built the fortress of Utumno underground and attacked before the Valar were prepared, destroying the lamps and wreaking havoc upon the world.
Growing afraid of Tulkas and Manwë, Melkor escaped to Utumno, and the Valar were unable to overcome him as their strength was occupied in repairing the damage he had caused.
Thus ended the Spring of Arda
And now for the longer, quote-y version. All quotes are from The Silmarillion by JRR Tolkien.
During the First War "a spirit of great strength and hardihood came to the aid of the Valar, hearing in the far heaven that there was battle in the Little Kingdom; and Arda as filled with the sound of his laughter. So came Tulkas the Strong, whose anger passes like a mighty wind, scattering the cloud and darkness before it..." and Melkor fled and forsook Arda and there was a long peace. Tulkas remained and became one of the Valar, and is especially hated by Melkor.
The Valar brought everything to order and "Yavanna planted at last the seeds that she had long devised. And since, when the fires were subdued or buried beneath the primeval hills, there was need of light, Aulë at the prayer of Yavanna wrought two mighty lamps for the lighting of the Middle-earth which he had built amid the encircling seas." Varda filled them with light, and they were set "upon high pillars, more lofty far than are any mountains of the later days." Illuin to the north of Middle-earth and Ormal to the south "and the light of the Lamps of the Valar flowed out over the Earth, so that all was lit as it were in a changeless day."
The seeds of Yavanna grew into a "multitude of growing things great and small" and animals came to live in the plains, rivers, and forests etc. In the midmost parts of the land the light of the lamps blended "and there upon the Isle of Almaren in the Great Lake was the first dwelling of the Valar when all things were young, and new-made green was yet a marvel in the eyes of the makers; and they were long content."
While the Valar rested, Manwë ordered a great feast "But Aulë and Tulkas were weary; for the craft of Aulë and the strength of Tulkas had been at the service of all without ceasing in the days of their labour." And Melkor, who still had spies among the Maiar, knew of all this. "Therefore he gathered to himself spirits out of the halls of Eä that he had perverted to his service" and he drew near to Arda.
The Valar gathered upon Almaren "fearing no evil, and because of the light of Illuin they did not perceive the shadow in the north that was cast from afar by Melkor, for he was grown as dark as the Night of the Void."
"And it is sung that in that feast of the Spring of Arda Tulkas espoused Nessa the sister of Oromë...Then Tulkas slept, being weary and content, and Melkor deemed that his hour had come." He came to Middle-earth in the north, unnoticed by the Valar. Melkor began building the fortress of Utumno "deep under the Earth, beneath dark mountains where the beams of Illuin were cold and dim."
Because of the evil of Melkor "Green things fell sick and rotted, and rivers were choked with weeds and slime, and fens were made, rank and poisonous, the breeding place of flies; and forests grew dark and perilous, the haunts of fear; and beasts became monsters of horn and ivory and dyed the earth with blood."
The Valar realized that Melkor had returned and sought for his location. Melkor came suddenly to war before the Valar were prepared and destroyed the lights of Illuin and Ormal.
"In the overthrow of the mighty pillars lands were broken and seas arose in tumult; and when the lamps were spilled destroying flame was poured out over the Earth. And the shape of arda and the symmetry of its waters and its lands was marred in that time, so that the first designs of the Valar were never after restored."
In the confusion "Melkor escaped, though fear fell upon him; for above the roaring of the seas he heard the voice of Manwë as a mighty wind, and the earth trembled beneath the feet of Tulkas." But he managed to reach Utumno before they caught him, and he hid there. The Valar could not stop him, for "the greater part of their strength was needed to restrain the tumults of the Earth".
Thus ended the Spring of Arda.
Notes:
Middle-earth
When Middle-earth is mentioned, it is not referring to the Middle-earth of the Hobbit or LOTR. So much destroying and rebuilding and moving of lands goes on before those books, that there is no relation to this first land, save the name. If it is confusing, blame Tolkien. In this section, I imagine Arda as being as sort of dark, watery blank slate, and then in the middle is this huge place with its own seas and islands (hence calling an earth and not a land), and somewhere in the middle of that is the Great Lake which holds the Isle of Almaren. I could be getting it all wrong, but it is so short lived that I am not terribly concerned, and I like to imagine it that way.
The First War
There is no particularly detailed account of the First War, but it is explained in Ainulindalë and Valaquenta as being rather long (though time wasn't really a thing yet) and consisting of the Valar building things and Melkor destroying them over and over again until, with the help of Tulkas, they finally cast him out.
Armies/Hosts
The hosts of the Valar and of Melkor are the Maiar.