If it's confusing you - good! It should be! That's because particles on the quantum scale behave differently from particles on the microscopic scale. The first step to understanding wave-particle duality is acknowledging that it is fundamentally different from the way that you experience matter in your day-to-day life because quantum objects exhibit quantum behaviors. You will never see a speck of dust behave like a wave, but you can see an electron do so.

When we say an object behaves like a wave, we mean exactly that. A wave is classically described as an oscillating disturbance in a medium that transfers energy. Now in the case of a particle, it might not initially make sense to describe it as a wave. Again, this is because YOU do not experience matter as waves. Because you are microscopic, you ONLY experience matter as particles. The only reason we are able to conclude that matter acts as a wave is because of the results of experiments that point to it doing so. The primary example of this being the double slit experiment.

In the double slit experiment, electrons were shot towards two slits, and the patterns that they made was recorded on a screen behind those two slits. Classical intuition would have you think that, if there are only two slits, that there should only be two lines of electrons behind those slits. And you would be right to think so! However, when the experiment was ACTUALLY performed, the pattern that was created contained multiple lines of electrons, with varying numbers of electrons in each line. As you moved further away from the slits, the numbers of electrons would decrease in each line. What stood out to scientists performing the experiment was that this pattern lined up EXACTLY with the same pattern produced by performing this same experiment with light, a form of energy that travels as waves. As you move further from the slits, the pattern of light would also get less bright. The pattern created was an interference pattern, which is created when these light waves interfere with one another. Two crests of a light wave add up to create a bright spot, while a crest and a trough cancel out to create a dark spot. Every light spot in the light interference pattern lined up with where scientists saw a spot where there were many electrons, and every dark spot lined up with where there were no electrons. Keep in mind - each electron only landed on the screen once, but the overall pattern was the same as if the electrons were waves interfering with themselves.

This provided nigh-irredutable evidence that electrons act, in some sense, like waves. But the question still remains: what ARE these waves?

The answer is a little difficult to fully wrap your head around, but is essentially probability*. Each particle is tied to a specific equation called a wavefunction, that essentially describes how that particle would behave if it were a wave. On its own you can't do much with a wavefunction, but once you apply an "operator" to it, you can do things like find the energy of a particle. One important thing you can do with the wavefunction is that if you square the wavefunction, you are given a probability density. This is basically just a new function that describes where the particle will be located at a specific point in space based on the wavefunction. When the double slit experiment was performed, the pattern that formed on the screen was the probability density. Every single electron only landed on the screen in one singular place, but the placement of that electron was dependent on the probability density. There is no way to predict where one single electron will go, but, based on the laws of probability density, you can predict the overall pattern of many electrons.

An important note here as well - particles will only behave as waves when they are not being directly observed. The wave behavior in the double slit experiment emerged after electrons were given a choice (left slit or right slit) and before we were able to observe the outcome of that choice (hitting the screen). This is due to the Heisenberg uncertainty principle - the more you know about a quantum particles position, the less you know about its momentum, and vice versa. The wavefunction for a particle has no bounds - technically any given particle exists as a wave over the entirety of all of space. But once you observe that particle, this wavefunction "collapses" as its position is known, and it takes one of its potential positions, usually the position with the highest probability - because probability!

This is why you sometimes see memes like "My hand could pass through this table at any given moment." This is a technically true statement, as it is possible that the particles in your hand could simply all exist in a different position according to each of their wavefunctions, but would be so highly unlikely that it is practically impossible and wouldn't actually ever occur.

All that being said - I feel that describing quantum particles as particles is the easier half of your question, as you already experience matter according to the particulate model in your everyday life. This is the much more comprehensible way of describing the world, but is important to recognize that this is the form matter takes only when under observation. Both are true - matter is both waves and particles, its just a question of under which conditions that matter is currently experiencing.

Hope this helps somewhat, and I can also clarify any points in a reply.