Economists define a public good as being non-rival and non-excludable. The non-rival part of this definition means that my consumption does not affect your consumption of a good; I do not "use it up". The non-excludable portion of this definition means that I cannot prevent you from consuming a good. Another way of understanding this concept is saying that adding an additional person to the public goods market has a marginal cost of $0. In other words, even those that do not explicitly (actually) pay for the good can benefit from the good.
While there may not be an example of a "pure public good", we can come up with a number of examples that may be close. National defense is one of the traditional examples; the idea here is that if the military is "protecting" Hank and Peggy, it is also protecting Dale, their neighbor that does not pay taxes. Another popular example is a lighthouse. While a port city may pay for its lighthouse by levying taxes on ships that come into its docks, it cannot charge ships that are passing by but still receive the navigational benefits of the lighthouse. It may help to look at a Goods Continium
To expand our understanding of public goods, let's look at how they are provided in the private sector, and then in the public sector.
We can also look at a More Advanced Discussion on Public Goods