How do atoms join together to form molecules?

Every electron has a charge which is of magnitude q=6.022*10^-19 C thus if two electrons were about r=10^-9 m apart(which is a very practical distance for two atoms approaching each other prior to a reaction) then the Energy of the system is F=Q^2/4*Ï€*ε*r where ε is the permittivity of the medium this energy comes out to be of value E=2.3*10^-19 N. But electrons in orbitals have 2 more characters namely spin and angular momentum,now angular momentum and/or spin induced dipoles are consideration that I wont take in hand since a force that is proportional to r^-6 is very negligible in comparison to the forces I am going to describe,otherwise so,but Ill return to them later. Now for every electron in an orbital the following are necessary- l and s where l is the angular momentum Quantum Number and s is spin Quantum number. now the force experienced by an electron in a orbital of angular quantum number  "l" is E=root((l*(l+1))*h^2/(2*Ï€^2*m*L^2) and for a spin of s the net spin angular moment is given by S=s1+s2,s1+s2-1.....s1-s2 then the net energy for such a spin is is E=root((s*(s+1))*h^2/(2*Ï€^2*m*L^2) for s=1/2 we have E=root(3/4)*h^2/(2*Ï€^2*9.1*10^-31*10^-18) the energy is allmost same as that of orbital energy the factor is E*2 all these energies lead to E(t)=3*E a net energy of E=2.36*10^-16 which is greater than the repulsive force between electrons! this means that this force plays a greater role in the Energetics of a bond formation rather than the repulsive forces themselves,so basically the bond forms if the net energy from spin and angular momentum of the final MO is lower than that of the initial MO that is why electrons tend to pair up with opposite spins since the square of E for a +1/2 spin is a positive quantity and that for a -1/2 spin is an equal and negative quantity thus they cancel each other and give rise to a much lower energy MO.This leads to the formation of a stable bond,since in this case spin angular moment,spin magnetic moment are mutually cancelled. This explanation is a more intuitive way of understanding the dynamics behind bond formation,but it should help while trying to understand how a bond is formed and how we see an electron in quantum universe(I like to call it that :D )

When two atoms are far apart there occurs no interaction between the molecules. When they come closer there occurs some repulsion(between the electrons of the different atom and two nuclei) as well as some attraction( between the nucleus of one atom and the electrons of the other atom). If the attraction predominates then a bond is formed (internuclear distance). If they get too close then again repulsion occurs.

This is one of those questions where just about any simple answer will have something wrong with it because there are a number of things going on, and any answer has to leap across some complicating aspects to avoid getting bogged down. The first part involves a fundamental aspect of physics that is not entirely agreed and lies within what interpretation of quantum mechanics you prefer. Chemistry assumes each electron occupies an orbital that is separable from the rest (which violates the state vector formalism, as I understand it, but chemistry is intractable unless this start is made. The good news is, if you make this assumption, an awful lot of chemistry becomes reasonably easily understood, and as far as I know there are no observations that falsify the assumption). There is a wave associated with the orbital (either a real wave, following de Broglie, Bohm, and me) or some mathematical construct. Such orbitals can contain one or two electrons (if two, generally with opposite spins) and may be bound by one or more nuclei. By bound, I mean each electron in the orbital experiences a positive field from a  nucleus. The simplest explanation is that wave interference means that electronic charge is increased due to wave interference between the nuclei, and this "glues" the nuclei together. Unfortunately, we can find a very few examples where this is seemingly wrong by observation, although in fairness it is probably reasonable in most cases. My personal view is the easiest way of looking at it is that prior to molecule formation, the waves are bound at infinity (i.e., reach zero amplitude). After bond formation the wave from atom A in the direction of B has its range decreased from infinity to somewhere where it gets dominated by  atom B. Decrease the positional range and the kinetic energy increases (from the increase in momentum due to the Uncertainty Principle) hence, from the virial theorem, which applies as long as one can consider the system has a Lagrangian, the total energy drops by an equal amount. So, the bond develops because positional range is decreased WITHOUT a repulsive force being generated between the atoms. If you wish to go a bit deeper, my next proposition is this. On the bond axis, the waves pair, and because there are now two electrons, we can consider the mass doubles. If it precisely doubles, then consider the action around each atom, and to shorten the discussion, use hydrogen as an example. The following is the simplest, and through being oversimplified is not quite correct, but it shows the major reason for the bond. If action is to remain quantised, which I believe underpins quantum mechanics, and the mass doubles on the bond axis, the covalent radius has to reduce to ao/√2, or 37.4 pm. The actual "half distance" is 37.1 pm. Because the frequency on the bond axis has doubled, due to doubling the number of electrons, the kinetic energy here doubles, as does the bond energy and hence is 1/3 the Rydberg energy, or about 437.4 kJ/mol, whereas the actual observed strength is 436 kJ/mol. That is oversimplified,  but it addresses the reason WHY there is a bond.

By sharing electrons. This is called covalent bonding. Viewed at a very basic level, atoms consist of a lump of positive charge surrounded by negative electrons. The positive nuclei repel each other, but this repulsion can be overcome by positioning electrons "between" them, generating an attractive force. This always happens with electron pairs: most often one coming from each of the atoms in the bond. See: https://en.wikipedia.org/wiki/Covalent_bond

Perfect! I just finished listening to a video on bonding in chemistry and this pops up! Covalent molecules require the sharing of electrons between two or more atoms of relatively equal electron affinity; i.e. between two nonmetals. While small differences in electronegativity will result in partial charges, these are extremely small. Ionic bonds, strictly speaking, do not constitute molecules. They make up compounds instead, held together by electrical interactions between charged ions. In ionic compounds, it is impossible to identify individual molecules, while it is possible in covalent molecules.

Electrons are the currency of trade for all bonding. They are of course attracted to their nucleus as well as that of the neighboring atom. Above you see calcium and notice too calcium only has two valence electrons in its valence (outermost) shell.  Chlorine  has 7 valence electrons and a slightly different nucleus with fewer + protons. When atoms come together their valence electrons do repel, no question. In quantum theory, this is overcome by their adopting spin up and reverse polar spin down configurations which somehow disrupt this repulsion (not a quantum expert). Furthermore more elements want octet configurations of electrons in their valence shells in order to be isoelectronic ("the same electron configuration") of their nearest noble gas and occasionally not the nearest noble gas but ALMOST always for s and p atoms the nearest noble gas. S and p orbital electrons ONLY are used to figure out if octet is achieved. So calcium will want to lose 2 electrons instead of gain 6 electrons to achieve octet while chlorine will want to gain one electron to be isoelectronic (content/stable). They do this by either sharing electrons in molecular (covalent) bonds or by exchanging (give and take) electrons in ionic bonds. Take calcium bonding with chlorine one wants to lose two the only gain one electron per chlorine so this will be ionic so both do just that however two chlorines are needed to catch all of calcium’s electrons donated. This is basically an all or none process of ionic bonding. In other words if the resulting compound is not neutral it mostly will not occur spontaneously. Of course being chemistry if you are asking as an engineer I have to be more detailed. CaCl2 is the resulting ionic compound and is held together by electrostatic attractions, + charges attract - charges. Ca is +2 and Cl is -1 apiece for a net compound charge of zero. Because the calcium and the chlorine have different sized nuclei and also different needs to solve their valence shell imbalances (octet), the electrons are shared unequally, creating polarity in the bond, unless the atoms bonding are identical or very similar in terms of their electronegativities the sharing (bonding) will be unequal.   Any book or website has this chart scaled from 1 to 4. The rule of thumb in bond description is that is the electronegativity differences are 0.5 or less the bond is nonpolar molecular, and  between 0.5 and 1.9 they are considered molecular but polarized toward the more electronegative atom. Finally, if the differences are greater than 1.9, the bond is considered ionic; one atom gets the electron(s) and the other loses them. Thus in ionic bonding one atom is formally negative, the other is formally positive. Now take chlorine bonding with chlorine. They both want to gain one, not possible. There is no way one atom’s nucleus can attract more than the other's because they are exactly the same and so are their electronegativities. In this case electrons are both shared molecularly (0 electronegativity difference).  These electrons are virtually equally shared by both chlorine atoms with no doubt some random polarity as they orbit around bonded atoms. For all intents and purposes, however, they form a completely non polar bond. In SCl6 there will be 6 polar covalent bonds. EN differences are a little over 0.5.

Of course electrons repel each other. But there are the positively charged nuclei also. The net result is that for many atoms, if they share electrons with their neighbors, they can move to a more stable, lower energy configuration. If the sharing is pretty equal, you have a covalent bond. If not, then it is ionic. BTW: that they can lower energy levels by sharing electrons is a quantum effect. That is why to understand the details of all the above, you need quantum mechanics, at least at the level of Schroedinger's non-relativistic equation without spin. Then you can show how the discrete energy levels result from eigenvalues of this equation. If you then make reasonable assumptions like LCAO, you can then go on to show that atomic orbitals can merge to form molecular orbitals and find the minimum energy. That is where the bonding takes place.

Not everything is molecules, molecules only form amongst non metallic elements in the periodic table which is the right hand "p" orbital block to the right, starting where you see Boron, but stopping short of the inert gases in the last column. Molecules are linked atom to atom via what is called a "covalent" bond. Co- means together and valent means relatively strong, like the word valiant, to be strong. So covalent means to be strong together. What happens is that atoms from this block have outer electrons that are already spatially localised  into non-spherical lobe-shaped orbitals. By occupying molecular orbitals instead of atomic orbitals, these electrons occupy a much larger volume of space, albeit still lobe shaped. Electrons with larger volume have lower kinetic energy. This means that electrons in these configurations, lobe shaped shaped orbitals between two atoms, have a lower overall energy. Even better, two electrons can occupy the same molecular orbital. The orbitals can be viewed as vibrational nodes, so a molecule is a stable pattern of interfering vibration formed when two atoms with an available electron each become associated. I should add that following symmetry principles, the lobes can be combined side on as well as "end to end" to create even more diffuse electronic ranges. The extreme case is when electrons become "degenerate" and delocalised over multiple atoms, which is called "aromaticity"  on account of this phenomenon first being encountered in molecules that were known to also have biological aromatic characteristics. This becomes even more relevant to metals incidently, where electrons completely lose track of which atom they belonged to.

Simple answer - when 2 or more atoms share a pair of electrons when their electron shells overlap. Complicated answer requires a lot of in-depth study - which I am not qualified to explain.