In special relativity the spatial distance between two events and the time between when they occur become things that depend on which observer you ask. Observers in different frames of reference will record different positions and times for events. So if the spacing and timing of events is no longer an objective fact, we have to ask what happens to our concepts of events happening in the future, past, or elsewhere (meaning in another place). The light cone is a concept that helps us divide up spacetime and understand what we can say about events in relativity.
We'll start this discussion by drawing a spacetime diagram, so you might want to check out that node first if you don't know about them yet. The light cone is drawn by choosing an event, which we usually just put at the center of our diagram, and then drawing two worldlines. One is a line that corresponds to a light ray moving in the positive x-direction, and one a light ray moving in the negative x-direction. So, each line should have a slope of 1, representing that it is moving with the speed of light in vacuum. I've done the best I can with ASCII art, so pretend those are straight lines with a slope of one. Here, then is the standard spacetime diagram for the light cone.
v/c=1 ct ^ v/c=1
\_ | _/
\_ future _/
\_ | _/
\_ | _/
\_ | _/
\_ | _/
elsewhere \_ | _/ elsewhere
\ | /
_/ | \_
_/ | \_
_/ | \_
_/ | \_
_/ | \_
_/ | \_
_/ past \_
/ | \
The two lines we've drawn make up the boundary of the light cone. It's called a light cone because it's made up by two lines representing light beams and because it draws out two cones (technically one, two naped cone). They'd be more like actual cones if we added another space dimension coming directly out of the monitor at you, in which case the set of all lines coming out of the origin with v=c would make a cone1.
The light cone divides up spacetime around the event in question into past, future, and "elsewhere". So let's talk about what that means. The important part to understanding these differentiations is the spacetime interval (sometimes also called the spacetime invariant). From the fact that all observers in special relativity measure the same speed of light one can argue that the the quantity
Δx2 - c2Δt2
is the same constant for all observers. Meaning that though the distance in space Δx and the length of time Δt between two events may be different for two different observers, they will both agree on the same value of the spacetime interval Δx2 - c2Δt2. Based on that fact we can start to reclaim some idea of past, future, and elsewhere.
Past and Future -- Timelike Separation
A point inside the light cone has |x| < c|t|. That means that x2 - c2t2 is negative. However, we've said that this is a number all observers can agree on, so that means that it's negative in all frames of reference. For the total to come out negative, we know that t' (the time for that event in some other frame of reference) can never be zero. If we know that t' never crosses zero, then it seems safe to assume2 that if t starts out greater than zero, it is greater than zero in all frames of reference, and if it's less than zero it's less than zero in all frames of reference; thus, we've justified our labels above. Something inside the future portion of the light cone happens after our event for all observers, so we can safely say it's in the future. An event in the past portion of the light cone happens before our event in every frame of reference, so we can safely say it's in the past. For a discussion about time ordering in relativity in more depth, see my writeup on relative simultaneity. Now, it should be noted that we can't say too much about whether an event inside the light cone happens at the same place in space as the origin or somewhere else. If all we need is x2 - c2t2 to be a negative value, we can find a frame of reference where x'= 0 and -c2t'2 is equal to the required constant. So for events inside the light cone there is always a frame of reference where the two events are coincident, meaning they happen at the same place. When one event falls within the light cone of another, and, thus, they have unambiguous order in time, the events are said to have a timelike separation.
Elsewhere -- Spacelike Separation
For a point outside the light cone |x| > c|t| and x2 - c2t2 is equal to a positive number. Now we can play the same trick and say that, since this must be the same positive number for all observers, x' cannot be zero in any frame of reference, since then x'2 - c2t'2 couldn't come out positive. We could then infer again that x doesn't cross zero, which is true with only one dimension of space but is not true with the three dimensions of space we actually have, so we won't focus on that3. Still, the fact that x' will never be zero in any frame of reference means that we can unambiguously say it is elsewhere than at the origin. We can't, however, say to much about when it is, because it is always possible to find a frame of reference in which t'=0 and x'2 has the positive value of the spacetime interval. When one event falls outside the light cone of the other, and the two events unambiguously happen in different places, they are said to have a spacelike separation.
On the Light Cone -- Lightlike Separation
There is one more class of points that we haven't discussed, and those are the points that actually fall on the light cone and have |x| = c|t|. Because the speed of light is the same for all observers, the worldlines that make up the light cone will be the same ones in every frame of reference, so a point that is on the light cone in one frame of reference is on the light cone in every frame of reference. Going back to the spacetime invariant, we can see that x2 - c2t2 = 0 for all the points on the light cone. If one event falls on the light cone of another, they are said to have a lightlike or null separation.
Though it is not a postulate of special relativity, it is generally accepted4 that no information or influence can propagate faster than the speed of light. This gives our definitions of past and future even more meaning. Since only events inside the future portion of the light cone can be connected to the origin by a worldline with a speed less than light, an event can only influence events that fall inside or on the boundary of the future portion of its light cone. For the same reasons only events in the past portion or the boundary of the light cone could have influenced the event at the origin. Thus, only events inside or on the boundary of the light cone are causally connected to the event at the origin. Furthermore, if the origin marks the position of a body moving though space, then it can only possibly move into the future region of the light cone and can only have come from the past region.
For completeness, I should also mention that the concept of a light cone also comes up in general relativity. In GR the light cone is a local concept that makes sense when you're talking about spacetime close to a particular event, since on a small scale spacetime looks fairly flat. The idea is basically the same, except that the spacetime interval that we've been using is replaced by distance according to the particular metric of that curved spacetime. The light cone is then the set of trajectories that give zero distance from the event in question, just as we've done above, and again it divides spacetime locally around that event into past, future, and elsewhere. One example in GR where light cones are often used is in the Schwarzschild geometry around a black hole. The light cones of events far away from the black hole look pretty much like the ones for Minkowski space that we analyzed above, but near the event horizon the light cones begin to tilt with the future light cone tilting more inward toward the black hole. Finally, at the event horizon the light cones are tilted so much that the entire future portion lies inside the event horizon; thus, an object at the event horizon can only go into the black hole as it does forward in time.
1 Imagine taking the picture above and rotating it around the time axis.
2 This follows from the fact that transformations between different frames of reference are continuous. This would generally be assumed from the outset, and if we know about the Lorentz transformations then we can readily show that it is so.
3 In more that one dimension of space we can easily make x' = -x just by choosing a frame of reference that is rotated 180 degrees around one of the other axes. To look at it another way, there are clearly many ways to go from x to -x without going through the origin in 3 dimensions. On the other hand, for events with timelike separation they are confined to stay inside the light cone, so they'd have to go through the origin.
4 It is accepted for good reason. Particles or information traveling faster than light introduces paradoxes into the theory and erodes the entire idea of causality. More importantly, no such phenomenon has ever been observed.
Note: The light cone is apparently also sometimes called an event cone or a null cone.
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.