What holds atoms together?
In non-nuclear chemistry, everything is electrostatic interactions. This is why you can learn and predict so much just by "following the electrons"


Covalent bonds are also formed because of electrostatic interactions - they are just more complicated conceptually than ionic (actually, ionic bonds are more accurately described by wavefunctions, we just try to keep things simple in the beginning). Since electrons exist as waves, when you confine them they start to do weird stuff (or what seems weird to us). For example, the shapes of atomic orbitals are all the way they are because the electrons act like standing, 3-dimensional waves trapped between a nucleus with a positive charge and a "zero-point" at infinite distance. When you put two atoms close together, the electrons on different atoms interact with each other, and the wavefunction becomes much more complicated. The result is that in some pairs of atoms, the wavefunctions combine to form bonding orbitals.


So the short answer to your first question is: "Molecular orbitals hold atoms together in covalent bonds, and those are a result of electrostatic interactions and the quantum nature of electrons."


Yes, ionic compounds are large collections of ions, and you can't really define "molecules" for them - instead we talk about "formula units" which are the lowest possible whole-number ratio of elements that represent the compound. Groups of covalently bonded atoms are also held together by electrostatic interactions, but since the covalent bonds are so much stronger, a molecular compound can exist "on its own" as a single molecule. Collectively, the forces that hold collections of molecules together are called van der Waals forces if they don't involve ions. In any atom or molecule, there is never a completely uniform charge density on the surface. For some molecules, this is extreme (water is a good example) and we say it is very polar, or that it has a large dipole moment. This is just another way of saying that one part has a negative charge and the other has a positive charge. In water it looks like this (from wikipedia):




In this picture, red means "more electrons" and blue means "less electrons." Water can form hydrogen bonds, which are very strong electrostatic interactions. Some atoms and molecules have an almost uniform charge density on the surface. We call these "non-polar" molecules - noble gases are good examples. However, even noble gases have what is called an induced dipole due to statistically correlated fluctuations in electron density when the atoms are near each other. As a result, even noble gases can be cooled to the point where they become liquid - the very, very weak electrostatic interactions will hold them together at low temperature, when they are not moving very fast. These forces are called London Dispersion Forces - after the guy who first described them. London dispersion forces are important, because they are found in all molecules - polar or not. In fact, this is what makes most plastics solid. Polyethylene, for example, is made of very long chains of essentially non-polar molecules (from wikipedia):



Each chain is attracted to the others through weak London dispersion forces, however, since each chain has tens of thousands of atoms, those tiny forces quickly add up to make large forces holding the polymer together. This is why polyethylene is solid at room temperature, and can be used to make things like shopping bags!