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Space is big. Really big. You just won't believe how vastly hugely mind-bogglingly big it is. You may think it's a long way down the road to the chemist, but that's just peanuts to space. See also this BBC Infographic attempting to convey distance just within the Solar System: How Big Is Our Own Solar System?
On the most basic level, "space" can be summed up as the X, Y, and Z (up/down, left/right, forward/backward) coordinates that make up the Universe. Space, generally, is everywhere that everything exists in. In some mathematical constructs and physical theories, space is melded together with time to make a four-dimensional spacetime. So where X, Y and Z are our physical dimensions, A to B represents the when we are inside of X, Y and Z.
"Space" is also used to refer to "outer space," or "up there as opposed to down here." There is no definite boundary where outer space begins, but it is generally said to begin about 80-100 kilometers above the surface of the Earth. This is just above the level of the mesopause (the top of the mesosphere) where temperatures are as low as -100° C and the atmospheric pressure is 0.2% of that at sea level, or less. The official U.S. definition of an astronaut is someone who has flown above this level.
Most of outer space is a near vacuum, that is, there are very few atoms or molecules floating around it (as little as one atom per cubic centimetre, which seems like a lot until you remember how small atoms are). It is mostly populated, apparently, by dark matter but contains a lot of dust and fine particles as well as objects that are large enough to be of concern: galaxies, stars, planets, asteroids, and so on.
"Space exploration" refers to the puny efforts of humanity to explore outer space. A lot of it is outside the scope of RationalWiki's mission, but more than a half century of space exploration has given birth to some pseudohistory and many, many conspiracy theories — from the Moon landing hoax accusations to the Face on Mars and beyond. Then there is the belief that private firms will now assume responsibility for opening the space frontier. That the state normally opens physical and technological frontiers is an argument made by both John Hickman and Neil deGrasse Tyson.