Proof-of-work is the oldest known means of securing a blockchain, still used by Bitcoin today. In this system, the right to confirm transactions (i.e. decide the order in which they are processed) goes to whomever is first to solve an arbitrarily-difficult math problem, such as finding an extremely small hash value. It also has other applications.
The most common criticism of proof-of-work is its energy usage and subsequent impact on the environment. Proponents however argue that most miners use electricity that is renewable or would otherwise be wasted. The best-known alternative is proof-of-stake.