This step in your academic career can be called multiple things: postdoctoral fellow, postdoctoral associate, postdoctoral scholar, postdoctoral researcher, or simply postdoc.
It is a position that you work in after obtaining a PhD and is considered the next step in the training to become an independent academic researcher. Dependent on where you are, you might be regarded as a full employee of the university or company or you may be considered a trainee, and still fall in the ‘student’ category when it comes to job status.
Postdoctoral fellows usually have their own projects and perform the research themselves, with help from students and other lab personnel. This job often requires you to first obtain funding before you can do your work, so grant writing is an important skill to learn at this stage of your career. You may also supervise undergrad and grad students in their projects and you might have a teaching affiliation with your university. Other tasks may include project management, lab management, negotiate with sales representatives, collaborate with researchers outside of your institution, and more.
Most postdoctoral positions are for 2 or 4 years and are mostly performed in the first 5 years after obtaining the PhD, but this is not a rule and there are many different ways a contract might look like. Someone can do multiple postdocs before getting a more permanent position or a move to a different career.