Blackholes are very massive things (maybe we can call them regions in our space or universe) from which even light can escape after one region (called event horizon). They are formed in nature if a star after many years (10^10 on average) run out of fuel (hydrogen which after doing nuclear fission liberates energy (which can be calculated using e=mc^2), then due to its own gravity collapse into a very dense, massive and sometimes even called as singularities. Now the question here is what will happen to Earth if the sun due to some mysterious power becomes a blackhole due to collapsing in its own gravity? (forget about any biological impact)
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I wouldn't say that nothing will happen to Earth, since without sunlight most life would cease to exist.
However, the only factors in determining the gravitational attraction between two objects are the masses of the objects and the distance between their centers of mass. Since neither of these factors are changing, the Earth (and other planets) would remain in their current orbits.