Typical examples are most roads, including the pavement, and public parks.
Typical differences between e.g. sitting on a public bench and sitting on a seat in a pavement café:
A public library is also more or less a public place, but some rules may apply which are absent outside.
In general, there is no expectation of privacy in a public space.
Public spaces are attractive for budget tourists and homeless people, especially those that are relatively comfortable, e.g. a shopping center that provides shelter and, in a cold climate, is heated (or cooled in a hot climate). Sometimes the presence of homeless people is not appreciated and measures are taken to make the public space less attractive to them; the comfort of regular users may be affected by these people but also by the measures against them, e.g. no benches, a lower temperature, waiting rooms that are locked in the evening, etc.
A shop is an example of what is intermediate between the two meanings: everybody can enter and look around without obligation to buy, but activities unrelated to the purpose of the shop are not unlimitedly permitted.