Well let's take a look at what that means..."public art." This is art that has been created with the intention of being displayed in a public place where it can be accessible to everyone. This, of course, contrasts with art that is privately held or is being kept in a museum. Sometimes this includes art inside of buildings that people can get to and see if they wanted to, but I think the spirit of true "public art" is that it is more accessible in that it is "out there."
This type of art has been around for a long time, with monuments and statues being some common forms people have been making for centuries. Some people also include in this definition examples of beautiful architecture which are meant to be enjoyed by everyone who sees them.
The art should not, also, be limited to physical things. Public art can be human demonstrations of various kinds, such as dances, parades, and other displays.
As for the purpose, well, it has the same purpose as any art: to be enjoyed, to make people think, or to convince people to believe a certain way. When taken too far, public art can become propaganda, but usually in society it functions as a way to beautify or express a common community value. The art becomes controversial when it "goes too far" or goes against those community values.
Public art, like its name suggests, is meant to be displayed where the public eye is. It is meant for everyone to view it, out in the open. Now for the function, why did the artist create such a work for anyone passing by to see it? Usually, artists do it to get their artwork known if they are an artist just beginning their career. Other artists like to display their work if it is about what is going on within their city/country. People use art to voice their opinions in a visual way. It also helps to get the public mind thinking.