Our sun can't turn into a black hole because, as Ancient Titan said, it doesn't have enough mass. But what no one has mentioned yet is how a black hole actually forms.
First you must have a star of sufficient mass (a supergiant)(our sun is only a red giant; a supergiant is 10 times bigger).
Then, when the star has used all it's gases and energy, it explodes in an blast called a supernova (you may have heard the term "going nova").
After this, one of two things can happen:
1. It can condense into an incredibly dense star. It is small and extremely heavy (a tiny speck would way as much as a house).
2. It can collapse into a black hole. If you fell into it, time would slow down for you and you would be stretched out like spaghetti.
Therefore, not all stars are large enough to go nova, and not all supernovas produce black holes.
Also, even if our sun did turn into a black hole (again, not enough mass) we would never feel the effects of it anyway because the supernova that precedes the black hole would vaporize us instantly, along several other planets near the sun! Cheers!