No, the stone around which Jerusalem's iconic Dome of the Rock was built is not a meteorite. The stone, which has been given so much significance by Muslims and Jews, is part of a simple bedrock limestone protrusion of the same rock that makes up the hill upon which Jerusalem is constructed. It does not stand alone as a singular rock.
Indeed, given all the interest about this outcropping, one would think that there is something special about the rock's composition, that perhaps it fell from space. The Muslim's believe it was from that rock that Mohammed ascended to heaven, and the Jews believe it was the place Abraham was told by God to sacrifice Isaac.
But perhaps it is fitting that there is nothing unusual about the rock at all. Rather, that it is a place set apart by the mind of man that gives it its special quality.
Your question doesn’t indicate what rock to which you are referring or where it might be, so we’ll focus on the part of your question where you mention how this rock looks different from other rocks. Geologists categorize rocks in three major groups depending upon what process formed them. Igneous rocks were formed from melted rock that cooled and formed solidified rocks. Sedimentary rocks are formed at the earth’s surface or in water and are comprised of layer upon layer of sediment. I believe each layer is indicative of another geological era. Metamorphic rocks are igneous or sedimentary rocks that are put under such extreme pressure or such extreme heat that their physical characteristics are completely changed. Thus you can see that there are probably many types of rocks with which you are not familiar.
One interesting rock specimen is Looking Glass Rock in Asheville, North Carolina. It is a pluton which is an intrusive igneous rock formed by crystallizing magma beneath the earth’s surface before it is thrust up out of the ground. These types of formations are extremely large as is Looking Glass Rock.