|MadSci Network: Astronomy|
Before we can answer this question, let's first clarify exactly what a black hole is. During a star's life, it undergoes nuclear fusion producing tremendous amounts of energy that we observe as heat and light. When a very large star exhausts its fuel, it becomes unstable and begins to collapse upon itself. The weight of this matter from all sides compressing towards the center creates a point of zero volume and infinite density. The resulting black hole has such a tremendous gravitiation field that not even light can escape.
Only very massive stars, those of more than three solar masses, can become black holes. Less massive stars, such as our Sun, become less compressed bodies such as a white dwarf. So, to answer your question, if our star were to implode it would not become a black hole because it is not massive enough.