The UFO is a miniature version of the Thing's UFO, it was constructed secretly by the now assimilated Blair underneath the tool shed using parts of machinery it could find, which included parts from MacReady's helicopter.
It was discovered by MacReady, Garry, and Nauls when they went to the tool shed to have Blair tested whether is he still human or not, once they reach the tool shed, Blair is nowhere to be found but they discovered a hole which leads to the UFO which is still under construction. Assuming that the Thing plans to hibernate in the UFO during the winter and plan to be found by any possible future rescue teams and finally infect the mainland, the three decided to make a last stand to stop the Thing once and for all by destroying the entire outpost, the three equipped themselves with dynamites and molotov cocktails and proceeded in destroying the entire outpost, MacReady later threw a stick of dynamite at the UFO, destroying it in the process.
The "Blair Monster" was somewhere in the area of 1-2 stories tall. Presuably the craft would be flown while still in human form/dimensions. During the Thing's final confrontation with Mac it obviously increased its bio-mass, surface area and volume to have the upper hand in the ensuing attack and unavoidably harsh outdoor climate. How it did so was by its ability to alter physique (and using the material of Garry, Nauls and the dead inert leftovers of Clark, Copper, Things etc.)
Exactly how it would "shrink" back down, if it intended to, or just detatch parts of itself to pilot the UFO is debatable. Possibly, both the mini UFO and the fullscale UFO would be used (assuming it was repairable) and the dog would be left to wander for good measure.