Neutron stars and black holes The official limits for a neutron star is $1.4 - 3.2\;M_\odot$. But I read that the limit depends on the particular structure of a star to estimate which mass it must have. I also read that neutron stars with less than $1.4\;M_\odot$ were observed. Given this information, I wonder if we can be sure that our Sun has definitely not enough mass to become a neutron star. Are there absolut limits (without the need of further information) for a star to become a neutron star or a black hole ?
 A: Observed neutron stars range from  $1.0 \pm 0.1 M_{\odot}$ to $2.7 \pm 0.2 M_{\odot}$ according to table 1 of  The Nuclear Equation of State and Neutron Star Masses, which lists dozens of examples. Keep in mind that the mass of the neutron star is typically substantially smaller than the mass of its progenitor star; late in the stellar life cycle a lot of mass is blown away, for instance a star that goes though an AGB phase may lose >50% of its mass. So our $1M_\odot$ Sun is likely to end up as a stellar remnant with $M < 1M_\odot$, probably a white dwarf.
According to Structure of Quark Stars, the mass is the only parameter to consider for neutron stars (but not hypothetical quark stars), although I would think rotation rate would be a factor. 
This reference also states that neutron stars can be as small as $0.1 M_{\odot}$, but this does not imply that the sun will actually become a neutron star.  
According to Possible ambiguities in the equation of state for neutron stars, it is the theory (equation of state) of neutron stars that is causing the current uncertainty about the limits of neutron stars.  
Also, it is unknown whether or not neutron stars may become quark stars before becoming black holes.  There is a term "quark nova" for such a hypothetical event.  
A: Yes, there are absolute limits (with some theoretical uncertainty) for the mass of a progenitor star that can become a neutron star or black hole and the Sun is well below that limit.
The other answers here talk about the range of masses of neutron stars, but do not directly answer the question you pose: the answer arises from considerations of what happens in the core of a star during the course of its evolution.
In a star of similar mass to the Sun, core hydrogen burning produces a helium ash. After about 10 billion years, the core is extinguished and hydrogen burning in a shell results in the production of a red giant. The red giant branch is terminated with the onset of core helium burning, leaving a core ash of carbon and oxygen via the triple alpha process. After the core is extinguished again, there is a complicated cycle of hydrogen and helium burning in shells around the core. During this phase, the star swells enormously to become an asymptotic red giant branch star (AGB). AGB stars are unstable to thermal pulsations and lose a large fraction of their envelopes via a massive wind. The Sun is expected to lose about $0.4-0.5M_{\odot}$ at this time.
Now we get to the crux of the answer. What is left behind is a core of carbon and oxygen, with maybe a thin layer of hydrogen/helium on top. With no nuclear reactions going on, this core contracts as far as it is able and cools. In a star governed by "normal" gas pressure, this process would continue until the centre was hot enough to ignite carbon and oxygen burning (a higher temperature is needed to overcome the greater Coulomb repulsion between more proton-rich nuclei). However, the cores of progenitor stars with masses $<8M_{\odot}$ are so dense that electron degeneracy pressure takes over. The electrons in the gas are compressed so much that the Pauli Exclusion Principle results in all the low energy states being filled completely, leaving many electrons with very high energies and momenta. It is this momentum that provides the pressure that supports the star. Crucially, this pressure is independent of temperature. This means that the core can continue to cool without contracting any further. As a result it does not get any hotter in the centre and fusion never restarts. The final fate of stars like the Sun, and anything with a main sequence mass of $<8M_{\odot}$ is to be a cooling white dwarf. The figure of $8M_{\odot}$ is uncertain by about $\pm 1M_{\odot}$, because the details of mass loss during the AGB phase are not completely solved theoretically and it is difficult to empirically estimate the progenitor masses of white dwarfs.
Stars more massive than this have cores which do contract sufficiently to begin further stages of fusion, resulting in the production of an iron/nickel core. Fusion cannot produce any more energy from these nuclei, which are at the peak of the binding energy per nucleon curve, and thus the star will ultimately collapse and has a core mass greater than can be supported by electron degeneracy pressure. It is this collapsing core which forms a neutron star or black hole.
An interesting caveat to my answer is that there may be an evolutionary route for a star like the Sun to become a neutron star if it were in a binary system. Accretion from a companion might increase the mass of the white dwarf star, pushing it above the Chandrasekhar mass - the maximum mass that can be supported by electron degeneracy pressure. Though in principle this might form a neutron star, it is considered that a more likely scenario is that the entire star will detonate as a Type Ia Supernova, leaving nothing behind. 
A: There are two questions here, namely about the limits on neutron star masses, and about the possibility of our sun becoming one. I'll try to argue that they are different questions, viz. the first about the stability and the second about the formation of such objects.
1) DavePhD's reference in the comments (here, for completeness) answers it completely. There is a lot of room for neutron star masses, because it depends intrinsically on the equation of state of nuclear (and possibly sub-nuclear) matter. Since we don't know the correct equation of state is hard to give strict boundaries. Without an equation of state one could have a mass as large as desired, just by increasing radius. So qualitatively the best one can do depends on the interplay between mass and radius, or density if you will. 
The strictest limit comes from Schwarzschild radius, that is if you make too dense a star it would generate an event horizon and collapse into a black hole. Next to this, one notes that the speed of sound escalates with density, so if you try to make too dense a star it will have speed of sound greater than the speed of light, violating causality. This gives a limitation in the different equations of state possible. The upper bounds of about 3.5 solar masses comes from this consideration. You'll find all this more deeply discussed in the aforementioned paper. The summary is in Figure 3, page 51. I am completely ignorant of an analogous argument for lower bounds on the masses that use only some physical principles (in spite of my first, incorrect, answer that related it to angular momentum and Rob Jeffries kindly corrected me on the comments) so I have deleted the incorrect previous part.
2)Somewhat independently of the previous discussion, we can be pretty sure that the sun will never become a neutron star, no matter what equation of state is correct. This is because gravitational collapse of a star is a highly non-linear process, that besides the different nuclear fusion cycles, will generate shock waves. Therefore it will not proceed adiabatically, on the contrary this processes will shed most of a star's mass. Therefore to produce a neutron star we need to start with a very heavy one, typically of the order of tens of solar masses. This is the reason we attribute neutron star formation to supernova events.
A: Black hole existence was predicted by solving Einstein’s general relativity equations. Mathematically, the equations show that it is possible to have a singularity at the center of a black hole.  The meaning of this singularity is that the mass of a black hole is confined to an infinitely small point at its center, thus the density at this point is infinite.  The main problem with this conclusion is that, at the singularity, the laws of physics don’t work. This is what makes the black hole a mystery and gives rise to mind-boggling theories such as that black holes are portals to other Universes, or that it is possible to travel in time.
I claim that although this solution is mathematically possible, it does not have a physical meaning. (Note: the same argument was also expressed by…  Einstein). I postulate that there must be a limit to the maximal density of bodies in the Universe. The current prevailing theory is as follows: A black hole is created when a star consumes its fuel and then gravitationally collapses. The end of this process is dependent on the mass of star. If the mass of the star is 1.39 solar masses (designated as Chandrasekar limit), gravity is strong enough to combine protons and electrons to make neutrons and thus creating a neutron star. The neutrons, and residual protons are packed in the neutron star at their maximum density. If the mass of the star is between 1.5 to 3 solar masses gravity becomes strong enough to break the nucleons into its constituents (quarks and gluons) and then the star becomes a black hole that has a singularity point at its center with an infinite density. 
Partially, I concur with the current theory.  Specifically:    1) The origin of a neutron star and a black hole is the gravitational collapse of a star.  2) The final mass of the neutron star or the black hole relates to the initial mass of the star.  Neutron stars have been observed in the Universe. The neutron star contains nucleons (neutrons and protons) that are packed to the maximum density possible in the Universe. The density of a neutron star is 3.7x10^17 to 5.9x10^17 kg/m^3, which is comparable to the approximate density of an atomic nucleus of 3x10^17 kg/m^3. The surface temperature of the neutron star is extremely high ~600000K.    https://en.wikipedia.org/wiki/Neutron_star 
As for the black hole:  I claim, that the mechanism that creates a neutron star is also applicable to a black hole. I mean that the creation of a black hole is not by compressing its mass further, thus breaking of nucleons into their fundamental constituents, as postulated by current theory, but rather by adding nucleons to the nucleus to get the maximum density.  There are two reasons for my claim. The first is theoretical, Pauli’s exclusion principle.  The exclusion principle forbids two identical fermion particles to occupy the same place at the same time. If the size of the black hole becomes infinitely small, then the nucleons must overlap each other, contrary to the Pauli’s exclusion principle.
The second reason is experimental. I note two known experiments.  The first experiment is measuring the force between nucleons as a function of distance between them.  In tests done in particles colliders, it was found that the force between two nucleons is as described in https://en.wikipedia.org/wiki/Nuclear_force.   In this graph, the force (in Newtons) is plotted against range - the distance between two nucleons (fm). The graph shows that for range smaller than 0.8fm, the force becomes large repulsive force. The conclusion is that two nucleons cannot be squeezed into the same space.
The second experiment was recently done by nuclear physicists at Jefferson Lab. They measured the distribution of pressure inside the proton.  The findings show that the proton’s building blocks, the quarks, are subjected to a pressure of 100 decillion Pascal (10^35) near the center of a proton, which is about 10 times greater than the pressure in the heart of a neutron star. This means that the outward-directed pressure from the center of the proton is greater than the inward-directed pressure near the proton’s periphery and therefore a neutron star cannot collapse.  https://www.jlab.org/node/7928
The question now is how come that black holes are not directly observed, while neutron stars are observed. My answer is:  The visibility depends on the relation between the physical radius of the nucleus and Schwarzschild radius. If a celestial body has a nucleus radius that is bigger than its Schwarzschild radius, it will be observed. On the other hand, if a celestial body has a nucleus radius that is smaller than its Schwarzschild radius, it will be hidden.
This is exemplified in the following calculations: 


The mass limit between a neutron star and a black hole.
There is a limit to the mass of a neutron star. At this limit, if more mass is added to the neutron star it will become a black hole. The limit mass can be found by equating the Schwarzschild radius to the radius the nucleus of the neutron star.
 
This result is in good agreement with observations. The smallest black hole observed in the Universe is XTE_J1650-500. Its mass is estimated to be ~5-10 Sun masses.
https://en.wikipedia.org/wiki/XTE_J1650-500
To sum up: 
1)  A black hole is basically a neutron star. Like a neutron star (and also the nucleus of an atom) it is compressed to the maximum density possible in the Universe.
2)  A black hole must have a mass that is bigger than ~5.25 Sun masses. At this mass the physical radius of the black hole is smaller than its Schwarzschild radius.
3)  It is possible that the temperature of the black hole is higher than the temperature of a neutron star. However, this temperature cannot be measured by an observer outside the Schwarzschild radius.
4)  It can be shown that the gravity of the neutron star and the gravity of the Milky Way’s black hole are of 2x10^12 m/sec^2 and 2.6x10^14 m/sec^2 respectfully.
5)  The physical conditions of 3) and 4) show that a black hole cannot be a portal to other Universes.
