Take a look at the first-quadrant graph on the right. On the horizontal axis, which we'll give the usual name of "\(\small\mathtt{x}\)," we'll measure distance in meters. But not just distance in one dimension (as when you measure how far a car has traveled away from you) but all 3 spatial dimensions.
Including more than one dimension on just one axis sounds complicated, but it really isn't. The car, for example, travels in all 3 dimensions. But when it stops, we can still run a straight line from ourselves--at the center of an imaginary sphere--to the vehicle to determine the distance it traveled. This distance is just a number (with a unit of measure).
Okay, on the vertical axis, we will measure time, so our 2D graph will represent all four dimensions--three spatial dimensions and one time dimension. What is represented, then, on that graph is called spacetime, which is a concept that, contrary to what might be popularly believed, was developed not by Albert Einstein, but by his one-time teacher Hermann Minkowski.
You can see that we give the name \(\small\mathtt{ct}\) to the vertical axis, and this is because there are two layers of interpretation to this axis. The first layer says that what is measured on the vertical axis is also in meters. That's true. Every increase of 1 on the vertical axis is just an increase of 1 meter. The second layer of interpretation says that the vertical axis measures the number of meters light can travel in 1 second. This is the "time" aspect of this axis.
I would suggest thinking of the value 1 on the \(\small\mathtt{ct}\) axis as being 1 "light-meter"--the distance in meters that light can travel in 1 second. But if you're desperate for a "time" unit, the 1 stands for \(\small\mathtt{\frac{1}{3 \cdot 10^8}}\) second, or 0.00000001 s. (That is, \(\small\mathtt{ct}\) really does mean \(\small\mathtt{c \times t}\): the speed of light, \(\small\mathtt{c = 3 \times 10^8}\) m/s, times the time, \(\small\mathtt{t}\), it takes light to travel a given number of meters.)
Now That We Have the Graph . . . and the Perspective . . .
What does a point mean on this graph? A point is an event--an instantaneous "happening" at a location \(\small\mathtt{x}\) and time \(\small\mathtt{ct}\) (though remember that an event actually has 4 coordinates \(\small\mathtt{x, y, z,}\) and \(\small\mathtt{ct}\)). And what about a line? On a spacetime diagram, we call lines world lines.
The picture shows the world line (world ray?) of a beam of light from a flashlight. There is an event indicated at (3, 2), but that event is pretty boring. It simply describes the beam of light at a spatial position of 3 meters and a time position of 2 "light-meters."
"Three meters from what?" is a good question. And we'll get to that. But we can also ask the interesting question, "What is the slope of the light beam?" The slope (\(\small\mathtt{\frac{ct}{x}}\)) of every light beam in a simple spacetime diagram is 1, because for light, covering a distance of 1 meter on the \(\small\mathtt{x}\)-axis always takes 1 "light-meter" of time on the \(\small\mathtt{ct}\)-axis. The speed of light (in a vacuum), according to the theory of special relativity, is a constant, so while its world line can be shifted left and right (and up and down?), its slope doesn't get any less steep than 1.
That Kid With the Flashlight Is Not the 'Observer'
What we see in the spacetime diagram above, though, is just one "inertial frame." For that inertial frame, we imagine an "observer" at the origin (0, 0). But, we can add another observer, as shown at the left, who is also stationed at (0, 0) relative to her own axes \(\small\mathtt{x'}\) and \(\small\mathtt{ct'}\) and to our first observer's axes \(\small\mathtt{x}\) and \(\small\mathtt{ct}\) (just ignore the kid with the flashlight). It can get pretty weird.
There are all kinds of interesting thoughts we can have and questions we can ask about this diagram.
For example, what does θ tell us?
What does tan(θ) mean? The \(\small\mathtt{x'}\)-axis is clearly not at a right angle to the \(\small\mathtt{ct'}\)-axis (relative to the first
inertial frame), and the red lines show how I constructed this axis. How is it done? Can two events occur simultaneously to one observer and at
different times to another? What does the graph of this look like?