The color of something you see is given by the electromagnetic radiation coming off of it. This means that in order that an object appear black, it must satisfy two properties:
- It must absorb (essentially) all the (visible) light that hits it, reflecting none of it back.
- It must be cool enough that its thermal glow is not visible.
Every body emits thermal radiation that depends on its temperature. However, for room temperature bodies, almost all the energy is radiated well into the infrared. On the other hand, we know that at temperatures of thousand of degrees, objects will glow visibly; that's what it means to be "red hot" (or hotter).
What it means to be "true" black is a linguistic question, not a physics question. Physicists tend to refer bodies that absorb all the radiation that hits them as "black," regardless of their temperature; so we can approximate the sun as a "black body" with a temperature of approximately 5700 K. However, that's just a choice of terminology; obviously the sun does not look black.
Sometimes, surfaces can have a black color but also be shiny. This is due to reflecting a relatively small amount of light (independent of color) and absorbing the rest. The way this is usually accomplished is by having a thin coating of partially reflective material over the underlying black pigmented surface that absorbs all the light that makes it through the surface layer.