The Arrowhead Project is a military experiment performed in Shaymore, apparently intended to create a trans-dimensional window with which to peer through into other dimensions. On July 17th, 1980 in the novella, and presumably in 2007 in the film, the experiment went awry (possibly due to damage from the Storm), and is believed to have released the Mist into our world.
Though it was never made certain in either the novella or the finished film whether or not the Arrowhead Project truly was what released the Mist and the creatures in it into our universe, the original script for the film included a prologue scene. In it, the scientists and military at the Arrowhead Project base successfully opened a portal to another dimension inside a tank.
However, lightning from the Storm struck the base, causing the portal to begin to expand and the Mist and the creatures began to pour through. Despite the Arrowhead Project personnel's attempts to close the portal, the creatures broke out of the tank which the portal was contained in, allowing the Mist to flood into the base and then presumably begin to spread across the country.