A conventional warp ship moves by changine the shape of the space time continuum around it, greating a gravity well in front of the ship and an antigravity well behind. The ship itself remains 'stationary' in a bubble of normal space, whilst the bubble moves at speeds thousands that of the speed of light.
The first Human warp flight was attained in the mid 21st century by Zephram Cochrane. Happily, unlike in the standard universe, World War 3 never occured, and the world simply created the United Nations Space Agency, which combined all of the Earths' existing space agencies into one. The experimental warp ship Quite Fast managed to attain warp 1 in 2065, allowing it to travel to the edge of the solar system in just a few minutes.
After this momentus occasion, the Earth rejoiced into a new age of civilization and cultures, and made its first step onto the galactic platform. Soon enough, humans had colonized Mars, the Moon, Alpha Centauri, and had made successful contact with the Vulcans.
By the early 22nd century, Humans were devoping the Constitution-Class starship, which entered humanity into a wider realm of exploration. The Constitution-class is referred to as 'Ship that Made the Federation'.
Alongside the 'Ship that Made the Federation', though, is the Constellation-Class starship developed in the mid to late 22nd century. The Constellation-Class has a motto of 'The Large, Clunky Van that Actually Built the Federation".
Humans entered the intergalactic playing field a few hundred years later than most other local species in the Alpha Quadrant, with the Vulcans, Andorians, Klingons and others all having used warp for at least 100 years prior to the Humans.