How exactly is fusion possible? It is a very dumb question but, I once read that fusion was possible when the wavefunction that describes the position of a particle overlaps with the wavefunction of other particle, but I think I don't fully understand this. Does this means that those particles have the same probable position? Do they have the same position? (I don't think they have the same position because of the Pauli's exclusion principle but I don't know much about this.)
 A: I am assuming you have had no rigorous mathematical exposure to quantum mechanics. Let me know if you do, and I can point you towards more specific material.
It is difficult to tell what it was exactly that you read, but here is my guess.
Why Fusion is Classically "Impossible"
There is Coulomb repulsion between two protons whose magnitude is:
$F=k\frac{e^2}{r^2}$ where $r$ is the distance between the two protons. This implies that the repulsion gets extremely strong as the two protons approach each other. You can think of this effectively as a "barrier" that the protons cannot penetrate easily. That is the electrostatic side of the story.
What actually causes nuclear fusion is strong nuclear interaction, which is a type of fundamental force with an extremely short effective range. That is, in order for two protons to start nuclear fusion, they will have to come extremely close to each other.
So, the the electrostatic repulsion makes fusion difficult. If the two protons were to get close enough to each other such that strong interaction triggers fusion, they will need "ridiculous" initial kinetic energy. Otherwise, they will get repelled away by Coulomb force before getting into the range of strong interactions. If you do the calculation (semi-)classically, the kinetic energy (temperature) required to penetrate the electrostatic barrier is a practically impossible amount. (By "classically," I mean "not taking quantum mechanics into account.)
Quantum Mechanics Saves the Day
Here is where quantum mechanics kicks in. Quantum mechanics describes particles with wavefunctions spanning a range of position (and other physical variables) instead of definite point-like positions.
Quantum wavefunctions often allow the particles to have non-zero probability of existing in the so-called "classically forbidden" region. What this means is that, even if a particle's energy may be too low to penetrate a energy barrier, quantum mechanics allows some finite probability of the particle appearing beyond the barrier. This is also called "quantum tunneling" effect. This is, I think, what is said to be "overlapping wavefunction" in the text you read.
So, at energy (temperature) much lower than what is required according to classical calculation, the protons have some (low but finite) probability of getting close enough so that strong interaction can trigger fusion process. Hence, nuclear fusion process begins at much lower, somewhat feasible temperature.
(I called the participating particles "protons" assuming a hydrogen-hydrogen collision, but the same principle is applicable to heavier nuclei.)
A: No, but you are on the right track. Fusion occurs when certain nuclei get close enough together that the short range nuclear force comes dominates the Coulomb repulsion. Remember that all nuclei have a positive charge in proportion to the number of protons in the nucleus. As two nuclei get very close together there is a strong Coulomb repulsion. This is one reason it is so hard to produce fusion reactions. If they do get close enough together for the nuclear force to come into play they can form different nuclei (elements). The difference in the binding energies between the initial and final nuclei is the energy released in a fusion reaction. One reason the Sun can maintain nuclear fusion is because the core of a star is at such a high density that the atoms/nuclei are close to each other allowing fusion to occur. And they are at such a high temperature that the motion of the nuclei produces collisions which bring the nuclei to the separation distance necessary for fusion to occur.
