For those who don't know what Black Hole actually is, here's a simple explanation. A black hole is a place in space where gravity pulls so much that even light can not get out. The gravity is so strong because matter has been squeezed into a tiny space. This can happen when a star is dying. Because no light can get out, people can't see black holes. They are invisible.
Why Sun cannot become a Black Hole?
Here's a reason. There is a limit, a minimum mass, which is required in order for gravity in a star that ran out of its nuclear fuel to cause that star to collapse into a black hole. This is known as the Tolman—Oppenheimer—Volkoff (TOV) limit, and it is about 2.2 times the mass of our Sun, according to our best present level of understanding.
That does not mean, of course, that every star heavier than this limit will collapse into a black hole. There are plenty of stars much heavier that will not. But it is the minimum mass required for gravitational collapse to occur, overcoming the so-called neutron degeneracy limit.
In fact, our Sun is not even heavy enough to collapse into a neutron star; its mass is below the Chandrasekhar limit of 1.4 solar masses, which is the minimum mass required for a neutron star to form.
The Future of our Sun and The End of Life
Our Sun will slowly increase in size and brightness over the next several billion years (eventually rendering the Earth uninhabitable). Then, near the end of its life, it will grow very large (may even swallow the Earth at this point) before it sheds its outer layers, with the remnants collapsing into a white dwarf. This dwarf star will no longer have nuclear processes producing internal heat, but it will be hot enough, dense enough, and compact enough to remain bright for a long time. But eventually it will cool down and become a dead remnant of the Sun. It will consist partially of degenerate matter, but it won’t be a neutron star and certainly not a black hole.
Never say never
If the universe lasts a very long time (and I mean really insanely long time, with the number of digits in the number of years exceeding a hundred trillion trillion, truly timescales that cannot be imagined in any meaningful sense of the word), the remnant of the Sun may, in fact, collapse into a black hole through quantum tunneling, since the black hole state is a lower energy state. Over time that black hole will evaporate through Hawking radiation; on these incredible timescales, that evaporation time is practically instantaneous (“only” sixty-some digits when measured in years, not a hundred trillion trillion), so this process really amounts to an incredibly improbable, fancy way of turning the remnant of the Sun into pure thermal radiation, dissipating into space.
The End Notes
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