A public joint stock company is a method to allow thousands or millions of people to jointly own a business. The most important feature is limited liability.
The most important function of a public joint stock company is that the investor can only lose their initial investment. Their liability is limited so that if the business fails they do not then have to pay more to cover any debts.
The stock of the company is the machines, plants, patents and so on and this is owned jointly. Each investor owns a small part of the whole. Any profits are divided out according to what share of the company each owns.
Public refers to ownership of a share of the company being open to the public. This most often means that it is traded on a stock exchange and anyone willing to pay the price on the day can become a part owner of the company.
Public joint stock companies are the way that the vast majority of the economy is organized. Everything you see about the New York Stock Exchange and every reference to stocks rising or falling is about people trading shares of public joint stock companies.
The combination of limited liability and widespread ownership is what enables large scale private sector businesses to exist. The public joint stock company is the defining structure of the modern American economy.