To put it simply, space is everything beyond the atmosphere of Earth. As for what it is made of, it is mainly emptiness, an almost perfect vacuum. "Almost" because there are actually lots of tiny charged particles (electrons, protons, and alpha particles [helium atoms without the electrons]) cast off from nearby stars like the Sun. In the case of our Sun, those charged particles flow outward from our Sun in what we call the solar wind.
Occasionally, one may encounter as a part of "space" other larger "bodies" (stars, planets, moons, asteroids, comets, meteoroids, dust, gas, and even a number of man-made spacecraft), but they are very far apart compared with their sizes. And somewhere out there, perhaps on a planet much like Earth circling around another star, there may be intelligent life!
Space has always been there, but it wasn't until 1957 that we actually sent spacecraft into space to measure what it is like. But asking who discovered space is a little like asking who first saw stars. The author Jules Verne wrote a book entitled From the Earth to the Moon in which he made some educated guesses about what space was like. Some he guessed pretty well; others he was way off base.
History suggests the Acadians probably were the first people to study space, but studies of the sky with telescopes started with Galileo in the early 1600s.
Stories of the adventures of Buck Rogers and others books that talked about humans exploring space in rockets raised our interest in space, but the Russian Sputnik in 1957 was the first craft to actually enter and stay in space for a long time. The Russians were also the first to launch human beings into orbit around the Earth. The USA was the first to send astronauts to another body in space (the Moon) and return them safely to Earth.
Very likely, the most exciting discoveries in space haven't been made yet. Perhaps someone who is in the second grade somewhere in the USA (or another country) will make some of those discoveries in the middle of the 21st Century!
In earlier times, a nebula was just any galaxy or cluster of stars outside the Milky Way galaxy (where we are). Now we use it to define collections of interstellar dust (interstellar means way out in space where the stars are) and gas visible as luminous patches or dark colors depending on how much dust or gas it has. Astronomers believe that these clouds of dust and gas are the birthplace of stars. There are different types of nebulas, some of which you can find at this website: http://www.seds.org/billa/twn/types.html.
A solar system is a tiny part of a galaxy.
Galaxies are star systems made up of hundreds of thousands or even billions of stars in addition to gas and dust. A solar system is a group of planets and other objects (comets, asteroids, etc.) in orbit around a star.
We call our star the Sun. Our solar system is located in a spiral arm of the Milky Way Galaxy. There are billions of galaxies and countless stars in the universe.
This link helps put it all in perspective: http://heasarc.gsfc.nasa.gov/docs/cosmic/cosmic.html.
As it stands now, there is no evidence of life beyond Earth. But it certainly can't be discounted as a possibility. The Deep Field picture taken by the Hubble telescope is littered with galaxies, each containing countless solar systems, and there are endless possibilities. You can see the picture here: http://antwrp.gsfc.nasa.gov/apod/ap960124.html. There is a debate over whether or not there is life outside of Earth. You can find a lot of great information about the debates and other updates on this subject at http://astrobiology.arc.nasa.gov/. A NASA program called Origins is also trying to answer the questions: Where do we come from? Are we alone? Over the next few decades, the program will work on building the telescopes and technologies that could help answer these questions.
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