To explain these, I am shamelessly going to use the (still) excellent cosmological figures produced by Tamara Davis. OK, let's start with this one.
Cartesian coordinate system, these are (x,y,z). As we are dealing with relativity, we are dealing with not only space, but space-time, and every point in the universe is labelled by 4 numbers, the three spatial coordinates and the time, t. So, every point is labelled as (t,x,y,z), and these points are called an Event.
The picture above is the evolution of our universe, the ΛCDM cosmological model, and has distance on the x-axis and time on the y-axis. The universe began at a finite time in the past (the Big Bang) and shows us (the vertical line in the centre) and other objects in the universe. At the Big Bang, distances between us and any other object is zero, and as the universe expands, objects move away from us.
The purple is the Hubble Sphere that we discussed last time. The question is, what are those other lines on there - the event horizon and the particle horizon? To answer this, we need to do a bit of mathemagic.
We need to note a couple of things. Firstly, the first picture is not the complete story, as we know that, if our current cosmology is correct, then while it was born a finite time in the past, it's going to last for ever. So really the first figure goes on for ever also. But we can fix that and come to it in a moment.
The x-axis actually shows "physical distance". Now, the proper distance is the multiplication of the Scale Factor (which depends on time and changes as the universe evolves) and what's known as the Comoving Coordinates (which, for any individual galaxy, are fixed values).
The Scale Factor's evolution depends upon what the universe is made of, and here's a few some people made earlier
This can give you a bit of a headache, but we know lots of functions that can map the infinite onto the finite (and mathematicians, don't complain about the terminology here :), such as tanh(x). So, when we look at our universe in terms of comoving coordinate and conformal time, we get the following
Now, the cool thing, the really cool think with these coordinates is that light rays travel at 45 degrees, not at the crazy curves we see above, and we see that our event horizon is made of such straight lines, meeting at where we head into the finite conformal infinity.
So, where does this get us? Well, the event horizon forms a triangle, separating events which are inside the triangle from those outside. Remembering that light rays travel at 45 degrees on this picture, and sit down with a pencil and a ruler, what you can see what this separation of events means.
If we pick an event within the triangle (remember, this is just a dot on this page) we can draw a light ray (travelling at 45 degrees) which hits us at the origin, somewhere between the Big Bang and the infinite future.
But if we choose an event outside of the event horizon and draw another light ray heading towards us, we will see that it will not be able to cross our path between the Big Bang and infinite future.
So, this means that the event horizon separates events from those that can ever send us a signal (i.e. we can see at some point in our history) from those that can't. The proper way of saying this is that the event horizon separates events into those that can have Causal Contact with us, from those that cannot.
This might seem weird, as if you think of a distant galaxy sending our light to us from a finite distance away, then, giving the fact that the universe will be infinitely old, we must receive the light at some point? But no, because the photon is battling the expansion of the universe, and may not win and we may never see it.
There is a flip side to this and if we take the above figure and extend the red light (our light cone) to the top of the picture, we can see something quite interesting. If we set off in a standard rocket, we can never travel faster than light, and so will be always within the future red triangle. What this means is that even though we have an infinite amount of time left to play in the universe, we can only explore this finite patch (as thing we are trying to get to are being pulled away from us by the cosmic expansion).
In fact, the longer we leave it, the less and less volume there is to explore! So we'd better head off right away if we are going to see anything!