The Standard Model is the term given to our representation of the structure of the atom. We understand that the atom is comprised of a nucleus composed of finer particles called neutrons and protons which have a neutral and positive charge respectively and that even smaller negatively charged particles called electrons whiz around this nucleus much like the planets orbit our sun. The electrons do this so fast as to make a determination of their position and momentum impossible for any point in time so we refer to their orbits, or energy levels as electron clouds and when they separate from the atom they become packets, or quanta of energy. We also know that some of these atomic particles are comprised of even smaller particles as in the case of protons and neutrons which are made up of different types of quarks. These quarks are said to be held together by particles called gluons. Quarks are further categorized and classified into more specific types and to extrapolate the constituents of those particles and of the leptons, which electrons are, falls into the realm of theoretical physics for the time being although a compelling attempt has been made to mathematically describe what are being called strings as part of a greater picture called M-theory, or Superstring theory. Simply put, it is presumed that these strings are the fundamental energy patterns of which the next higher level which are the particles are comprised. At the penultimate level of scale, which we shall call sub-cosmic, we find our own reality. We can describe ourselves as an ordered organization of different types of cells made up of certain molecules which are in turn assemblages of atoms. It is not inappropriate for the reader to consider each nuclear particle as analogous to a unique wooden Chinese puzzle ball, although that is a bit of a stretch.
The behavior of the atom is now well understood. The original arrangement of what we call the periodic table of the elements is generally accredited to Dmitri Mendeleev in 1869. The science of chemistry is an empirical field of study and our description of the fundamental nature of matter is the result of much research and experimentation which support theories and by which we have verified our conclusions. This leads us to the very important question, "how do atoms work?". One might even consider that question an invitation to evade a more fundamental one which is, "what holds atoms together?" or, "what keeps atoms from flying apart?".
"Nature abhors a vacuum." - so said Aristotle. While his statement actually referred to the workings of the lever-operated water pump it also holds true for the realm of quantum physics in the form of the Casimir force. This is the force which is described as that which is responsible for holding atoms together and refers to the fluctuations of the energy fields in the empty space between and around objects. It is not an electromagnetic force or a gravity force and it is measurable only at the extremes of proximity.
If we scaled an atom's nucleus to the size of a golf ball then the outermost electron orbital would be about 12 kilometers away. One might question that since there is seemingly much more empty space between subatomic particles than there is space occupied by those particles then why don't objects fall apart? How can any object be solid?
Atoms cannot mesh because of the strong inter-atomic forces which bind the electrons to the oppositely charged protons inside the nucleus. The known subatomic particles have their properties well defined and they exhibit characteristics which allow physicists to identify them. Insomuch as all matter is made up of energy, that energy of which these particles are made have definite limits in their normal state and they exhibit properties which confine those characteristics and which make them identifiable and to some extent predictable, notwithstanding Heisenberg's uncertainty principle. In a sense, we are describing a law of quantum mechanics: that particles can only respond - and correspond - to discrete values of energies.
The space between and around particles also contains fluctuating energies but those energies possess no discrete boundaries which might confine them and potentially define a particle nature. Those energies are said to occur as virtual particles which randomly pop into and out of the vacuum of the space giving the vacuum an average zero point energy. Another law of quantum mechanics which is analogous to Heisenberg's uncertainty principle which forbids knowing the position and momentum of a particle forbids knowing both the energy and time of the electromagnetic wave properties of a particle and therefore the absolute value of a system cannot be defined as zero. A vacuum can therefore not exist as a still and inactive void but only as a quantum state of matter and energy fields.
In the presence of two flat surfaces brought very close together the fluctuating virtual particles exert a greater pressure outside the plates than between them, therefore causing them to be attracted to one another. This observed phenomenon is known as the Casimir effect and was predicted by Hendrik B. G. Casimir in 1948.
So much for why atoms don't fly apart. That does not effectively explain how atoms work, only the role that the Casimir effect plays in keeping atoms together, notwithstanding that that is a very important role.
To understand how atoms actually 'work' we would need to fully understand the as yet hypothetical Theory of Everything, also known as the Grand Unified Theory. We can assume that there is a symbiosis of sorts to all that exists, that insomuch that the particles of matter, the atoms, can be represented as waves that those waves are always interacting with one another. The unique properties of each element can be defined by the specific wave natures of their atomic particles, and in that their interaction is at the same time constructive and destructive as a function of the waves of the similar and dissimilar atoms with which they interact. Therein lies the foundation for the discrete realms of quantum thresholds which allow definition of the particle, atomic, and molecular states. What they do is what they do and if things were any different then THAT is the way that things would be - the same with HOW they do it.