Any way of getting from one place to another using something which you don't own.
This might be a local scheduled bus, a cable-car, a taxi, a ferry, an aeroplane, a rental car - or even a hired bicycle.

But there are other answers.
Some people think that "public transport" is transport that is subsidised - or even totally paid-for - by the government.
Other think that it's transport which people share with other people. So scheduled aeroplanes count - but taxis don't. Wikipedia supports this approach, saying:
Public transport (North American English: public transportation or public transit) is a shared passenger transport service which is available for use by the general public, as distinct from modes such as taxicab, carpooling or hired buses which are not shared by strangers without private arrangement.
The New Zealand government thinks that public transport is:
A service for the carriage of passengers for hire or reward by means of a large passenger service vehicle; a small passenger service vehicle; a ferry; a hovercraft; a rail vehicle; or any other mode of transport (other than air transport) that runs to a schedule and is available to the public generally. It does not include:Ref: Guidelines for Regional Public Transport Plans, 2013 - glossary
- A taxi service;
- A dial-a-driver service;
- A shuttle service;
- An ambulance service;
- A private hire service
- A service:
- that is contracted or funded by the Ministry of Education for the purpose of transporting school children to and from school
- carrying passengers that is operated to transport all those passengers to a predetermined event
- that is operated primarily for the purpose of providing a tourism experience, rather than for transporting people from place to place.