What are examples of cost-push inflation?

The canonical answer is an increase in the price of oil.

Oil is used as an input in so many production processes, and we use fuel to transport ourselves and various goods.

So, when the price of oil goes up, it results in increases in production costs, and this tends to lead to price increases in a number of other goods and services – i.e. it leads to inflation.

This is cost-push inflation (as opposed to demand-pull) because it originated from the supply-side: it was an increase in the cost of production that led to the inflation.

Many other examples of cost-push are possible. Unions demanding (and receiving) higher wages for workers is perhaps another good example. (Since labour is also a widely-used input, the reasoning is similar to that of an oil price increase.)