A black body need not be of black color. Black body means a body which ABSORBS all wavelengths completely.
This gives rise to two conclusions:
- A black body doesn't reflect any light. A mirror can't be a black body because when you shine a light beam on a mirror, it doesn't absorb it completely, because it reflects some wavelengths.
- A black body can't be a transparent object. Because when you shine a beam of light on a transparent object, it doesn't absorb it, instead it allows some wavelengths to go through it.
Now, consider a situation where you have a black body in the sunshine. According to our definition, it will absorb all the wavelengths coming from the sun onto it. Saying that it absorbs those wavelengths, it means that it is absorbing their energy. According to Kirchhoff's law, a body having some finite temperature will emit ALL wavelengths of radiation, but the wavelengths themselves can be of different intensities like red is being emitted more intensely than the blue one. The wavelength which is emitted most intensely will be the color that body and I will call that wavelength "max intense wavelength". Our black body lying in the sunshine will also emit radiation of all kinds of wavelengths with different intensities and there would be a wavelength which would be emitted most intensely and that will determine the color of the black-body.
If that max intense wavelength lies in the visible region then the black-body will have some visible color like blue or red.
But when the max intense wavelength doesn't lie in the visible region, because we see only the visible range of the Electromagnetic spectrum, we will see only that wavelength which has maximum intensity in the visible region, though the overall maximum of intensity is out of range of visible light.
When the max intense wavelength is far far away from the visible region then the visible wavelengths will have very low intensities and hence no color is visible and black-body looks black.
For a black-body to look white, it will have to emit wavelengths corresponding to the visible region with nearly equal intensities because white light is composed of visible colors but with EQUAL intensities of all the colors.
Our sun has its max intense wavelength lying in the visible region and human eyes are evolved to see the most intense wavelength which is visible light.
In our situation of black-body absorbing radiation from the sun, it is radiating some energy on its own because of its finite temperature. So it is receiving some power from the sun and emitting some power on its own. When the temperature of black-body becomes constant then it means that power incoming will be equal to power outgoing.
If your black paint absorbs all wavelengths completely then only it is a black-body.