First off all, you would typically not go straight from finishing a PhD to a professorship. Unless one is extraordinarily brilliant and did groundbreaking doctoral research, several of years of work as a "post-doctoral researcher" (aka a "post-doc") are typically required before having a reasonable shot at a tenure-stream professorship.
Professorships are typically research positions. There are positions at most universities to work exclusively as a teacher, but such people are usually called lecturers rather than professors (though this is starting to change). However, the majority of professorships are research stream. At a North American university, you would usually start as an assistant professor (untenured) when, on top of your lecturing duties (which tend to be the smallest component of a prof's work) you work to produce more original research. At this point, you would generally have the assistance of graduate students. If you do good work as an assistant professor—generally over the course of 5 years, give or take a few—you get tenure (ie a permanent job) and promoted to associate professor. As an associate professor, you continue to do research but the day-to-day work is mostly done by your graduate students. So, you more or less complete the transition to being a managerial position in your group's research. In some cases, this transition may have already happened at the assistant prof level. There are, of course, exceptions with some professors who prefer to stay very hands on late into their careers—but this is broadly what happens. Now, even as a manager you are still very much actively involved in the research. You're responsible for its over all direction, for getting grant money to fund it, for overseeing the results, etc. In other words, still very much a working scientist. For many professors, this is the terminal position for their career. Those are very good—or, at the very least, long-lived if the university just goes by seniority—will at some point be promoted to "full professor". Aside from the prestige and probably a raise, there isn't much difference when going from associate to full professor.
Meanwhile, as profs gain experience and seniority they will often be expected to take on various administrative roles, like serving as a department chair for a few years or the faculty dean, vice-dean, etc. They will generally continue their research roles through such appointments, though at a reduced capacity. If they don't have such an administrative position, they will often have to sit on various committees like for thesis examinations or graduate admissions. And, yes, they will usually be teaching undergraduates along the way.
At universities outside North America, the process is largely the same though some of the formalities and the names of the various positions might be different. Wherever you are, professors will almost always be working scientists—though, as I said, the trend is towards a more managerial role over the course of a career—and are very, very busy people.