A public good is non-excludable (you can't stop non-payers from using it) and non-rival (one person's use doesn't diminish another's). Classic examples: national defense, basic research, clean air.
The problem: if you can't exclude non-payers, why would anyone pay? Free-rider incentives push provision below the socially optimal level. Markets underprovide public goods.
The solutions are all imperfect: government provision (how do you know optimal quantity?), private provision with bundling (radio + advertising), norms and social pressure. None generalizes cleanly.
Related: commons-and-tragedy, coordination-problems