Step 1: The idea behind variable oxidation states. In d-block metals the last electrons enter the \((n-1)d\) shell while the \(ns\) shell is already filled. These two shells lie almost at the same energy level, so there is no sharp jump in ionisation energy between removing an \(ns\) electron and removing a \((n-1)d\) electron.
Step 2: Because of this, a transition metal can give up different numbers of electrons under different conditions, producing a range of oxidation states that usually differ by one unit. Manganese, for instance, ranges from \(+2\) in \(\text{MnO}\) to \(+7\) in \(\text{KMnO}_4\). This is why they show variable valency.
Step 3: Copper as a transition metal. The rule is simple: an element is a transition metal if it has a partly filled d sub-shell in the elemental state or in any of its usual ions.
Step 4: Neutral copper is \(3d^{10}4s^{1}\) and \(\text{Cu}^{+}\) is \(3d^{10}\) (both full). However, the well known cupric ion \(\text{Cu}^{2+}\) is \(3d^{9}\), a partly filled d sub-shell. Compounds such as \(\text{CuSO}_4\) and \(\text{CuO}\) contain this ion.
Step 5: Therefore, because at least one common ion of copper has an incomplete d-orbital, copper is correctly grouped with the transition metals.
\[\boxed{\text{Partly filled } d \text{ in } \text{Cu}^{2+} \Rightarrow \text{transition element}}\]