The mind directly perceives sights, sounds, smells, tastes, feelings, thoughts. Some thoughts are ideas or theories connecting perceptions into meaningful patterns, which can then be labeled by a word. Ideas can be labeled by the mind as true or untrue. An idea labeled as true could be called a belief. The collection of ideas labeled as true is labeled as reality.
Drawing a distinction between true and untrue, and between real and unreal, seems to be related to two things: first, a desire that expectations of perceptions be in harmony with perceptions that actually arise; and second, a belief that perceptions have an origin in something other than perception.
If an idea leads to an expectation that is met, the idea is labeled "true," and if an idea leads to an expectation that is not met, the idea is labeled "untrue." Over time, the human mind compiles an ever-larger assembly of ideas refined by this process. The human perceives a certain compatibility among these ideas, so that they form a whole which is labeled reality.
The idea of reality contains many interesting contradictions. The first is that the concept of "reality" is itself merely an idea, or compatibility of ideas, but it claims to be something outside, separate from and superior to ideas. If this claim to superiority is believed, then a contradiction follows: "This particular idea labeled reality is not an idea."
As another example of contradictions, some particular definition of reality might include two ideas: First, that chairs are distinct objects of solid substance; second, that modern science is an accurate understanding of how things are constituted. The contradiction is that modern science has demonstrated that chairs, like everything else, are mostly (or possibly entirely) empty space, that there are no boundaries distinguishing the chair from its surrounding environment, and that if anything is there, it is either energetically swirling energy or incredibly tiny vibrating multidimensional membranes.
The reason the contradiction is not normally exposed is that both ideas are approximations, and each "works" in its own arena. That is, each explains certain perceptions. The everyday idea of "chair" leads to expectations that match perceptions in everyday life. The scientific idea of "chair" leads to expectations that match perceptions in scientific laboratories. Neither one perfectly explains all our perceptions.
If one accepts the idea that all our ideas are appromixations, incomplete theories of how perceptions will behave, then the ideas of "truth" and "reality" also fall into this category. "Truth" and "reality" are not ironclad absolutes outside us, but approximations, attempts to explain our perceptions. Believing with certainty that "reality" corresponds to something outside ideas is not justifiable by any idea.
That said, perceptions do generally seem to behave according to patterns, and the idea of "reality" could be seen as an attempt to identify these patterns. But how are patterns identified? What assumptions are made at the root of this "reality" concept?
The mind uses certain tools to build and link a chain of ideas. Logic is one means of testing ideas for compatibility -- to see whether an idea can be labeled "true". But how can logic itself be labeled "true"? Certainly not with logic. It must either be an assumption -- a leap of faith -- or merely a recognition that perceptions generally are in harmony with logic's predictions.
There are many such assumptions that the mind makes quietly, without questioning: concepts like space and time, self and other. All perceptions are filtered through these assumptions before being presented to the conscious mind as knowledge. Anyone interested in fundamental questions about knowledge and reality will have to look for and examine these assumptions as critically as possible.
For Further Exploration
The philosophically minded and curious can browse these sources elsewhere on the Web:
Immanuel Kant's Critique of Pure Reason: Wikipedia entry (including links to various translations) on one of the leading works of epistemology and metaphysics of all time; Kant addresses the distinction between the form of knowledge (pre-existing assumptions contained within the nature of the mind itself) and the content of knowledge (contributed by the world)