So, there has been quite a bit of talk about fields but then there also appeared a particle, the photon. And both have been associated with electromagnetism. But what is it, really?
Well, this question baffled scientists in the early 20th century. There was a lot of talk about a particle-wave-duality and things like that, which are still used as a simple explanation that things are either like a wave of like a particle, depending on the circumstances. And wave is connected to field, because a field is like an ocean: The height of the water at each point is also a kind of field. And like an ocean, there can be waves on it.
All that sounds a bit confusing? Indeed, people have made up their minds by now. And despite the usefulness of the picture of something which can be either particle or wave it is rather that it is both simultaneously. And the thing connecting it is the field.
Go back to the analogy with the ocean. Imagine that your field is an ocean. If the ocean is totally flat, there are no ripples and nothing else, so you could say that there is nothing happening. That is what people call a vacuum when they talk about fields: Just a field where each point looks exactly the same as everywhere else, and there is no change from one point to another.
Now, imagine, something is happening. Whenever something happens in an ocean, it makes ripples and finally waves. That is what people call an excitation of a field. Something is moving. Now, when you are very close by, then you just see the waves around you, and they do not have much of a structure. They are just waves. On the other side, if you are very far away then what happens just looks like a point, or a flat ball. That is exactly the analogy to the question whether it is particle or wave. If what happens (the 'excitation') is very far away, you do not see an internal structure to it, it is like a point. If this would be beneath the surface, it would look like a ball. And that is what you are usually refereeing to as a particle. If you go closer and closer, then the internal structure becomes apparent, and you see that the thing is much more like a wave again, rather than a particle.
Of course, this analogy can only be approximate. Just think of a moving particle: That would be like all the waves stay together and move at as a whole. You usually do not see this on an ocean - that what was originally a particle dissolves into waves, never to reunite again. That is different for the fields in the standard model. They can keep together, and even come together again if they have resolved earlier. One should keep these limits in mind when working with such analogies that they have their limits.
Anyway, sticking with the analogy, it is possible to see another important concept. If you are far away than the average distance between two peaks of the waves is very small compared to your distance. On the other hand, when you are close, the distance between two peaks is of similar order as your distance. This tells you that the relative sizes are important if you want to resolve the internal workings of something. You need to have something which is of the same size as the internal structure of the thing you want to analyze.
Particles are very tiny (the proton is of size 0,00000000000001 meters, the electron to the best of our knowledge smaller than 0,000000000000000000001 meters!). If you want to investigate their inner workings, you will need something which is even smaller. The only thing which is smaller than a particle is another particle. And there is also something else, which comes to help - it is possible to make a particle effectively small by making it faster. That sounds a bit weird, but it is not so far off. Think of the following: Take a parking car. Mark its beginning and end by going first to the front, and place a marker. Then walk to the end of the car, and when you reach it, put another marker. Measure the distance between both markers. Now try the same when the car moves. If you walk with the same speed, you will not get as far as when the car stood still, because it moves. It appears shorter, smaller. Now that may appear as cheating, and in a sense it is. But the laws of nature actually make this cheating true, by a much more subtle mechanism, called special relativity. This is a topic of its known, to which I will return in due time. For the moment, the only important thing is that if you want to probe a particle with another particle of the same kind, you need to make the probe particle move very fast compared to the particle you wish to analyze. That is the reason to build particle accelerators: Their only purpose is to get very fast particles to probe very short distances. And this is in fact not a simple task, and requires the most modern technology available to date.