A fair coin means it has proper cylindrical shape with narrow width and two faces which are equal in all respects. The coin in proper in geometry and uniform in density. When the coin is tossed there is an equal chance of getting a head or a tail. Suppose the fair coin is tossed (a very large number) 1,000,000 times, then 500,000 times the head turns up and 500,000 times the tail shows up.
An unbiased die means every face of the die from 1 to 6 has an equal chance of showing up on top. That means the die has a uniform cubical shape and density. It has all faces marked with exactly one number from 1 to 6 on it.
For example, if the shape is such that the face on which 4 is marked, is wider than other faces, then it will have higher possibility of being on the base. So the die will not then be unbiased.