Black hole's gravity Is gravity relative to volume, or size of an object? 

Since a black hole is a massive star that collapses on its weight, how comes the same sun's mass, when it becomes a black hole, provides gravity that disallows even light to come out of the black hole? Before it was a black hole, the sun had a similar weight to the black hole, since no mass was lost or gained (and particles, including photons, come out of the star). Once it collapses under its weight, does the gravity become more in relation to the size of the object in question (reduction from the huge surface area of a massive star, to a compact/compressed, low surface area black hole), now that the same non-increased or subtracted gravity is bound to a much smaller surface? 

Or in other words, what is it that makes everything bound to a black hole, but nothing of the same happens to a sun of similar mass, (and therefore with the same gravitational pull).
A Black Hole (BH) is an object of General Relativity (GR), not of Newtonian physics, so the answer involves both.

First Newton: As another answer notes, the acceleration due to gravity depends on the mass of the body divided by the square of the distance from it.  At a given distance (say, ten million miles) the acceleration due to gravity is simply proportional to the central body's mass.  It doesn't matter if the mass is iron, water, stone or chunky-style peanut butter. It's all exactly the same as long as you're far enough away to be outside it all.

(A GR sidelight: This is even true of BHs as long as you're more than a few BH radii away from the BH.  The gravitational field of a BH becomes Newtonian fairly quickly as you move away from it and, further out, a BH and a equal mass of chunky-style peanut butter have the same gravitational field.  If an object -- any object -- collapses into a BH without losing or gaining mass, its gravitational field at a reasonable distance away remains the same.)

Still Newton: The surface gravity of an object depends on the size of the object.  If an object is denser, it will be smaller for the same mass and its surface gravity will be higher. This is due to the division by the square of the distance. For a denser object, the surface is closer to its center and the surface gravity is accordingly higher.

(This is why the Moon's surface gravity is 1/6 Earth's in spite of the Moon being only 1.2% of the Earth's mass. It's also 1/4 the diameter of the Earth, so at the Lunar surface you are 4 times closer to the center and thus its mass affects you 16 times as much.)

Black Holes are General Relativistic objects, and the differences from old-fashioned Newtonian objects show up when you get within a few radii of the BH.  The effects that close are subtle and complex and not much like the kinds of forces we're familiar with.  Basically (handwaving really fast) the geometry of space-time becomes more and more distorted as you approach the BH's surface and at the Event Horizon, the geometry is such that there is no longer a direction out -- all directions lead in towards the center. (This is what defines the Event Horizon, actually.)

So the main thing that differentiates an ordinary mass from a BH, is that in a BH the object's mass is so concentrated that time and geometry themselves are distorted to the point that here is no way out.

This is fundamentally different from the simple increase in the gravitational field's strength that Newton's theory predicts.