What is the difference between atoms, particles and matter?

I would go with:

Particles are point-like pieces of matter. Depending on what you are discussing, it makes sense to regard elementary particles (those we do not know have parts, like electrons and quarks), composite particles (they have parts, like protons) or even atoms or dust (or in astronomy, even planets and stars) as particles. But the key thing is (1) it is localized and "small", (2) it does not have, or we do not care about, parts.

Atoms consist of electrons, protons and neutrons bound together by the electromagnetic and strong nuclear force.

Now, matter is the tricky part. A common definition is "anything that has mass and volume", and most matter we meet is atoms. But black holes have mass and volume, and free electrons have mass but may not have any volume (being, perhaps, properly point-like particles).

Particle physicists would say that what gives matter its tendency to take up space is that it consists of fermions, those elementary particles that have half-integer spin: the details are irrelevant, but stuff with fermions (like electrons, protons and quarks) cannot be pushed together arbitrarily like light and non-fermions can. So they would define matter as "everything composed of elementary fermions". (Where that puts black holes and dark matter can be debated.)

When people use terms, the meaning often depends on what they talk about. It is often useful to just check what definition they want to use, or state one oneself to make the discussion clear. It is rare that terms have a perfect, unambiguous meaning.

To summarize and simplify the other answer.

The word "particle" can have different meanings. In physics, when we talk about particles we usually talk about electrons, protons, neutrons. Atoms are made of a central nucleus and one or more electrons. The nucleus is made of protons and neutrons. The simplest atom is the hydrogen atom, made of one proton (nucleus) and one electron.

Matter is a collective term which refers to everything that is not light. Particles, atoms, and everything made of atoms, for example molecules, gases, liquids, solids, all of these things is called "matter".