Reaction of hydrogen peroxide and catalase
Manganese (IV) oxide, iron (III) chloride, and lead (IV) oxide are common examples for a catalyst that isn't found inside living organisms and which can break down $\ce{H2O2}$. Potassium iodide can also be used as a catalyst, along with other transition metal compounds, of metals such as silver or platinum. 

This didn't take much effort to find so I'm sure you can find more examples easily.

As for catalase, it seems to just decompose $\ce{H2O2}$, but this is useful in many different areas. For example, decomposing $\ce{H2O2}$ in contact lens cleaning solutions, preventing foods from being oxidised in wrappers (in conjunction with glucose oxidase, another enzyme), and removing $\ce{H2O2}$ from milk before cheese is made. The decomposition reaction is also used as a test for catalase in bacteria.

Sources: General use - National Center of Biotechnology Education - and more uses (including the cheese example) - Worthington Enzyme Manual