in basic terms market failure is caused in a perfectly competetive  markets through different channels:

some examples are

1. price regulation by the government (price floor and price ceiling)

an example is a rent ceiling in housing markets.

2. taxes and subsidies

3. externalities

4. Market power: In a perfectly competitive market we assume that there are large number of buyers and sellers and no single buyer or seller can influence the market price. However, in the real world there are markets with a single seller (monopoly) that can control prices. In such cases market outcome will not be efficient.

I don;t know what kind of examples you are looking for at what level but these examples are from intro econ course.

hope this helps