We do not know the exact shape of space, (I am assuming you are talking about the universe as a whole) as we cannot see all of it. The farther you look away from Earth, the fainter and smaller everything becomes. A star's red shift also makes things far away hard to see. We can only see the observable universe, which for all we know could only be a very small portion of the universe. We do know that the universe is not infinite though. The fact that the sky is dark at night shows this. If the universe was infinite, stars would be everywhere you looked in the night sky, as eventually, trillions of light years from Earth, there would have to be a star to fill up every point in the sky you can see. Since there are giant gaps between the stars in the night sky, this shows that the universe is finite.
Current models of the universe suggest that the universe came from the "big bang." 380,000 years after the big bang, a process known as inflation occurred, which expanded our universe to (almost) the size it is today. Hubble proved that the universe is still expanding, 13+ billion years after inflation. This has been (somewhat) proved by observing Cosmic Background Radiation, (CMB) radiation left from inflation. The big bang is thought of as a random occurrence, occurring only because, eventually, something had to happen.
We are in one of the "arms" of the Milky Way Galaxy.