Which type of reaction is #2Cu(NO_3)_2 -> 2CuO + 4NO_2 + O_2#?

The reaction given to you describes the thermal decomposition of copper(II) nitrate, #"Cu"("NO"_3)_2#.

In general, you can recognize a decomposition reaction by the fact that you have one reactant and multiple products.

Simply put, a decomposition reaction features one chemical compound being broken up to form other chemical species.

In this case, anhydrous copper(II) nitrate undergoes thermal decomposition to form oxygen gas, #"O"_2#, nitrogen dioxide, #"NO"_2#, and copper(II) oxide, #"CuO"#.

Here's a video that shows the thermal decomposition of copper(II) nitrate trihydrate, #"Cu"("NO"_3)_2 * 3"H"_2"O"#.

The only difference between the thermal decomposition of #"Cu"("NO"_3)_2# and #"Cu"("NO"_3)_2 * 3"H"_2"O"# lies with the fact that the hydrate will first dissolve in its water of crystalization.

Once the water of crystalization is completely evaporated, the decomposition of the hydrate is practically identical to that of the anhydrous salt.

The black residue that you can see in the end of the video is copper(II) oxide, the solid that's left behind after nitrogen dioxide and oxygen gas are given off.