This is a question that is easy to ask, but can be difficult to answer. In order to answer we need to understand what it means to say that we know something. This study is called Epistemology. If you're not up for reading a long Wikipedia article, this comic also offers a pretty good explanation of epistemology.
If you'd read either of the two links you would know the most common answer. Knowledge is, "Justified, true belief." When we say that we know x, we mean that we have good reason to believe that x is true. There are some problems with that answer, but it's a good start.
The nice thing about this definition is that it's simple. We all know what beliefs are, and I think we all know what truth is. The interesting question is, what counts as a justification? How do we know that our beliefs are justified? That's the question I'm interested in answering.
First of all, there are some beliefs that seem to be self evident. These beliefs don't have a clear justification. These beliefs are our basic assumptions about how the world works. Some of these beliefs can be almost invisible. They're so basic that it doesn't even occur to us that they might be questioned.
For example, most of the time we assume that our memories and our sense perceptions are accurate. We know that at times both can be deceived, but under normal circumstances we don't need to provide an explanation for why our senses and our memories can be trusted.
Another example is logic. If you've taken a logic course, then you've probably been taught that logical arguments are truth-preserving. Most people accept that that is true without the need for any further justification. Indeed, it's hard to imagine how someone would provide further justification for that belief.
The interesting thing about this first class of beliefs is, just because something seems self evident to us doesn't mean that it actually is self evident. If you and everyone around you shares a similar belief system then it is natural to assume that those beliefs are self evident. For this reason I think it's good to interact with people with radically different beliefs. They keep us honest and force us to examine our beliefs.
Of course, most of our knowledge isn't self evident. Most of our beliefs require some form of justification. Loosely speaking, these justifications can come in one of two forms.
First of all, some of our beliefs are justified logically. We start with those beliefs that we consider to be self evident and we use some form of logical reasoning to discover additional truths.
Secondly, some of our beliefs are justified by experience. Assuming that we can trust our perceptions and our memories they can be a rich source of knowledge.
In practice, most of the things we know represent a synthesis of our basic assumptions, our reasoning ability and our perceptions. All of these factors come into play when we talk about what we know.
The problem is that these are complicated issues and we don't all address them the same way. The result is that two different people can end up having very different beliefs, even if they're both very smart and they both think things through very carefully.
What about you? What assumptions do you make about the world? What roles do reason and experience play in shaping your beliefs?