Ashort-term job or work project that can be paid or unpaid and can lead to a full-time, paying position is called a(n)

internship
Internship is an example of opportunity given by employers to prospective employees that are currently in their undergraduate years. They will be working in the company for a specific period of time and maybe hired by the company if performance is approved by the management.  Internship is also known as work placement that can be full-time or even part-time but it is usually part time when interns are from the university and during semestral periods while it can be full-time during summer periods.