Why is copper diamagnetic?

What is missing here ist the relation to physical/chemical status of that "copper". What are You talking about? Copper metal? Copper atoms as vapour? Copper ions I or II in aqueous solution? As a matter of fact, copper atoms are paramagnetic (one unpaired electron is enough despite the number of the paired ones!) BTW, the Herren Stern and Gerlach knew that silver atoms are paramagnetic, guess why!. In bulk copper metal the odd electron is sent into the pool of electrons making the metallic bond, thus the metal is diamagnetic, the same is for Cu+ salts, whreas Cu++ salts are paramagnetic.

As Lubos has already told, the magnetic property originates from the extent to which electrons are filled in different orbitals.

If you have a look at the periodic table arrangement of d-block elements, you can see that Copper, Silver and Gold are on the same group due to their similar configuration $(n-1)d^{10}\ ns^1$. I don't know whether gold is diamagnetic (My periodic table shows: no data for Gold's magnetic properties). But from the look of copper and silver, gold should be diamagnetic too. These elements have still got an unpaired electron. So, they are paramagnetic. But it's so negligible that their diamagnetic property is enhanced.

It's probably determined by the magnetic moment $\mu=\sqrt{n(n+1)}\ \text{BM}$ (Bohr Magneton) where $n$ is the number of unpaired electrons. Hence, the more number of unpaired electrons an element has, the more paramagnetic character it has.