An atom is a thing; an element is a type of thing.
An atom is a collection of protons, neutrons and electrons. A single, isolated atom in its neutral state has some number of protons, the same number of electrons and some number neutrons (about as many as protons for the lighter elements, up to about 50% more, for heavier elements). The number of neutrons or protons in an atom only changes as a result of radioactive processes or very high-energy interactions such as you get in particle accelerators. And I mean really high energy: even if you think about blowing up sticks of dynamite, that isn't nearly enough energy to start messing with the protons and neutrons. Chemistry happens when atoms come together and share electrons or give electrons to each other. Chemical reactions happen all the time and many of them don't need a whole lot of energy: moving electrons from atom to atom is often very easy.
So, the chemistry of an atom depends on the number of electrons and the number of electrons in an isolated atom depends directly on the number of protons. Electrons are so easy to add and remove from atoms (just rub a balloon against your hair: the static electricity is because you transferred electrons between your hair and the balloon) so we classify atoms according to the number of protons in them. Neutrons aren't so relevant: I'll put a note about them at the end.
So, the element of an atom is determined by the number of protons. All atoms of hydrogen have one proton, and all atoms with one proton are hydrogen. Two protons is helium, three is lithium, seventeen is chlorine, 79 is gold, etc. A pure sample of an element contains only atoms of that type: for example, a pure sample of iron contains only atoms with 26 protons. Water, on the other hand, is not an element: a molecule of water consists of two hydrogen atoms (one proton each) sharing electrons with an oxygen atom (eight protons).
Now, what does it mean to claim that an element "cannot be broken down into a simpler form," and why aren't atoms "a simpler form"? Well, they're not a simpler form because an iron atom is iron: it's the same form, not simpler. Think of it this way. If I give you a lump of pure iron, all you can do is break it into smaller lumps of iron, or make it into a more complex substance, for example by letting it rust – rust is formed of iron and oxygen. The smallest possible lump of iron you could make is a single iron atom, but that's still just an unbelievably tiny lump of iron. If you wanted to break down a lump of iron beyond single iron atoms, you'd need to use a nuclear reactor or a particle accelerator or something like that and then, finally, you'd be able to get something that wasn't iron, because you'd have changed the number of protons in the atoms.
Let's contrast that with water. If I give you a bucket of pure water then, just like the lump of iron, you can divide it into smaller and smaller samples, ultimately ending up with a single water molecule. But you can do something else: if you pass electricity through the water, it splits into pure hydrogen and pure oxygen. These are "simpler" substances because each consists of atoms of just one element, whereas water has atoms of two elements.
What about neutrons? Well, in terms of chemistry, they don't do a whole lot and atoms that have the same number of protons but different numbers of neutrons are much more alike (they have essentially the same chemistry, for example) than atoms that have the same number of neutrons but different numbers of protons. It makes much more sense to classify by the number of protons, since that determines the number of electrons and that determines the chemistry.
Suppose you tried instead to classify atoms according to the number of neutrons. Well, most argon atoms (18 protons) have 22 neutrons, but some chlorine atoms (17 protons) and a decent fraction of potassium atoms (19 protons) also have 22 neutrons. As you probably know, argon, chlorine and potassium are absolutely nothing like each other. On the other hand, potassium atoms with 22 neutrons behave almost identically to the most common kind of potassium atoms, which have 21 neutrons.