Material from the star's surface will star to flow onto the black hole. At this stage it may become large enough for its outer shells to be gravitationally attracted by the black hole short distance away.
For a brief period of time (it is brief from the point of view of the age of the star, but it may last millions of years), before it explodes as a supernova too, it will grow up in size and lower its surface temperature. The other star, assuming it has been able to survive the cataclysmic explosion of its sister, will soon be getting to the end of its live too. When that happens, it will explode as a supernova and likely create a black hole from what was left from its core.
The heavier star will evolve slightly more rapidly and will run out of fuel sooner. Imagine a binary system containing two heavy stars of quite similar but still different masses. Our Sun makes quite an exception here because only every fifth star in our Galaxy lives on its own without a companion. Most of the stars live in pairs as binaries or in some more complicated systems. It means that the most massive black hole of stellar origin can have a mass of about few hundred solar masses too. The most massive stars have masses only up to about 500 solar masses otherwise they are not stable. This guarantees that after all the nuclear fuel has been consumed and the star explodes and casts off outer shells, there is still enough mass in its core with enough gravity to collapse into a black hole. The star has to have mass at least ten times larger than is the mass of Sun.
The stellar black holes arise as a final evolution stage of massive stars. Then there are supermassive black holes with masses greater than million solar masses. There are stellar mass black holes that have masses from few to at most several tens of the mass of Sun. However, that is way into the future and right now, there is more danger that we will cause our own destruction (by wars and pollution) than anything happening due to the Sun.We recognize two major classes of black holes based on their mass. At that point, Earth will become so hot that there will no longer be any oceans and will probably be the end of life as we know it. In another 4.5 billion years (that is a long time), the Sun will expand into a "red giant" star extending beyond the Earth's orbit. However, note that we are not safe on Earth forever. However, as there will be no longer any sunlight, we will have trouble keeping warm. Black holes do not go about scooping up people and planets. Now, to answer your second question: If hypothetically the Sun suddenly becomes a black hole (which will not happen), then nothing will happen to the Earth. So, do not worry about the Sun becoming a black hole. Stars much more massive than the Sun explode as a supernova leaving behind either a "neutron star" or a "black hole". Stars end their lives in two different ways: those will mass around the mass of the Sun will end their lives in a gentle way, becoming a planetary nebula and leaving behind a remnant called "white dwarf".
The Sun will never burst and will not become a black hole. Now will the sun burst one day? If it bursts what will happen to life on earth? I read that when stars burst it becomes a black hole.
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