In short: The particle horizon is the extent of our future light cone at $t=0$, and the event horizon is the extent of our past light cone at $t=\infty$. It is important to be clear that these horizons are horizons only to an observer at this place in Space and Time. They do not mark physical boundaries between different regions in Space; rather, they describe boundaries for regions in Space and Time observable to us, on Earth, at this particular time. An observer somewhere in the Virgo cluster will see different horizons, spherical, at the same distances as we see them, and with themselves in the exact centre.
This is the distance from which light emitted at t=0 can have reached us today. In all practicality, this is marked by the Cosmic Microwave Background at a redshift $z \approx 1100$, before which the Universe was opaque, so we cannot see further back when looking at electromagnetic radiation. There is hope, though, that neutrino and Gravitational Wave observatories in the not too far future can push us closer to the $z = \infty$ limit at the Big Bang. The Particle horizon is always expanding, in all cosmological models. In our currently favoured cosmological model, it is around 43 billion light years away.
This is the distance, inside of which photons emitted in our direction today will reach us in a finite time in the future. When light in distant parts of the Universe is emitted towards us, their journey is a race between their velocity towards us and the cosmological expansion of Space. The expansion of Space is homogeneous, which means that the recession velocity of a region is proportional to its distance. There is a distance, at which the recession velocity equals the speed of light; this is the distance we call the Hubble Sphere. However, in most cosmological models, the Hubble Sphere is not a horizon; it is perfectly possible to observe galaxies that are currently and have always been receding from us at superluminal speed; we have been doing this routinely for decades. This is because even though these photons are initially moving away from us, they are moving into regions of Space which are receding more slowly from us, until the "catch up" with our Hubble Sphere and start approaching us.
However, if the expansion is fast enough, it will "win" over the expansion of the Hubble sphere at some distance. This means that beyond this distance, photons will never get to our Hubble Sphere and thus never reach our telescopes, because the Universe expands so rapidly that the "crawl" towards our cosmic neighbourhood is too slow. Unlike the Particle Horizon, not all cosmological models have an event horizon - but the one which seems to describe our Universe does, and it is at a redshift of around $z\sim 2$.
Yes, we routinely observe objects at distances far beyond that of the event horizon. The Event Horizon is a horizon in Time, not only in Space. What we observe at those distances is all in the past, though. As the Universe expands, these objects cross our event horizon at a given time in their history. From Earth, when we look at a galaxy at these distances, we will see the light from this galaxy being increasingly redshifted as we observe epochs closer and closer to this point in time. This also means a slowing down of the perceived time; on top of the increasing redshift, we will see the galaxy's history slow down and grind to a complete halt as we come asymptotically closer to this point in its history - this event - beyond which we can never get more information. We can easily observe galaxies at $z = 2$, but what happens at these galaxies right now, we will never be able to know.
Remember that an event in relativity is a point in Spacetime. the Event Horizon is aptly named - it is not a geometrical horizon made up of points in Space, but made up of events; of points in Spacetime.