In special relativity the distance between where two events happen and the time between when they occur are not things that all different observers can agree on. For observers in different frames of reference the distance and time between two events will be different, so the concept of distance or time alone is not as meaningful as it was before we knew about relativity. However, Einstein found that there are certain things that do stay the same for all observers when we view space and time together as spacetime. One way of understanding some of the things that are going on in relativity is by making diagrams of spacetime, usually called just a spacetime diagram, or sometimes a Minkowski diagram.
The basics of spacetime diagrams
For simplicity, we'll draw a spacetime diagram for one dimension of space and one dimension of time. Of course, the real world has 3 dimensions of space, but together with time that would require a 4-dimensional diagram, and that is hard to draw. In order to make a diagram of spacetime, we'll put time on the vertical axis and space on the horizontal axis. Since we want to consider space and time as one concept, it helps if we measure them in similar units that make the comparison natural. Usually what we do is to make our time axis ct, that's time multiplied by the speed of light in vacuum. If time is measured in seconds and c is in meters per second, then out time axis will have units of meters, just like our space axis1. Now, let's look at a sample spacetime diagram.
A B ct ^ D E
\_ \ | / _/
\_ \ | / _/
\_ \ | / _/
\_ \ | / _/
\_ \ | / _/ . a
\_ \ | / _/
\_\ | /_/
\\|//
-------------------+-------------------> x
_//|\\_
_/ / | \ \_
_/ / | \ \_
_/ / | \ \_
_/ / | \ \_
_/ / | \ \_
_/ / | .b \ \_
_/ / | \ \
V
I know that's pretty crude, but try to bear with me. I've tried to draw four lines on the diagram, labeled A, B, D, and E, as well as two points labeled a, and b. A single point on the diagram represents an event, something that happens at a specific place and a specific time. Examples would be a car accident, a lightning strike, saying hello to your friend, or hitting a ball with a baseball bat. A, B, D, and E are each an example of a worldline, which is many events joined together to form a curve. For example, a body moving through space draws out a worldline. At each instant it is at a certain place and time, which is an event. All the worldlines above are supposed to be straight. I did the best I could with ASCII art. Worldlines don't have to be straight, but we're just going to talk about straight ones for now.
The slope of a worldline tells you something about the velocity of the particle it describes. Remember that velocity is x/t (for something that's not accelerating), so on this strange graph the slope of a line is ct/x = c/v. You could also say that 1/slope = v/c. Something that is stationary only advances in time, so it's worldline is vertical. The faster a body is moving, the further it's worldline is sloped out toward the x-axis. The worldline for something moving at the speed of light will have a slope ct/x = c/c = 1. That's why we chose specifically to measure time as ct. Worldlines A and E are intended to be straight, with slope one, so those are the v=c worldlines and might correspond to light rays. Worldlines B and D have some velocity less than c and greater than zero and could represent massive bodies moving though space. We could also draw a line with a slope of less than one, but since that worldline corresponds to a velocity greater than the speed of light it won't represent the motion of an actual object.
The uses of spacetime diagrams
I told you that position and time are different for different observers, and an event is defined by a position and a time. So the point representing an event will be in a different place if we draw a spacetime diagram in terms of the coordinates of another frame of reference; however, the spacetime diagram gives us a way to visualize what's going on. For one thing, guided by the Lorentz transformations we can draw another set of axes, showing the coordinates of another frame of reference and begin to understand the relationship between the different systems of coordinates. But that's a subject for another node. The worldlines will generally have a different slope in another frame of reference, representing the new velocity in that frame of reference, except a line having a slope of one, because the speed of light is the same in all frames of reference. We can also use the spacetime diagram to get some sense of the meaning of relationships in space and time in relativity, as I discuss in the light cone node.
Using the spacetime diagram we can begin to go beyond algebra and start thinking of special relativity in terms of geometry. Even though we've drawn it on an ordinary 2 dimensional plane, people generally don't think of this as Euclidean geometry, because in special relativity it isn't the ordinary distance in the plane that is important but the spacetime interval. This amounts to a new, strange way of measuring distances that defines a new sort of geometry, which is discussed somewhat under the spacetime node. This sort of space is called a Minkowski space. While this is not exactly the way that Einstein originally conceived special relativity, it is a very useful way of understanding it and is generally considered a more modern approach. Einstein did later use these concepts of spacetime geometry when we created general relativity, in which we start with a space like the one above and curve it in order to describe non-inertial frames of reference and the effects of gravity.
1 We could just as well make the time axis t and the space axis x/c, in which case we might measure x in terms of light seconds or light years.
Sources: My own knowledge of special relativity.
Though I didn't really consult any sources to write this, if you want to look at some, here are some suggestions:
The book I originally learned from (not necessarily recommended)
A. P. French, Special Relativity
A well respected introduction to special relativity from the view of geometry (spacetime diagrams) at an introductory/intermediate college level
Taylor and Wheeler, Spacetime Physics
Finally, most general physics text books will have a chapter somewhere near the back about relativity, which generally will contain a section on this subject.