What's a Professor of Practice do? Are they tenured / on tenure track?

Are they more like clinical teaching professors?

Do they perform research? 

I recently saw this job with lots of teaching duties but no research listed.
This answer will be a little different from that of JoshuaZ. 

Yes, such a position is intended for people wanting to build a career as teachers/educators, rather than theoretical researchers. But there are caveats. 

First, the job may not come with a tenure track leading to eventual traditional tenure. Instead, as at Duke, after a probationary period the holders get long term (say ten year) contracts that are renewable. The ones I am familiar with also come with various ranks. You don't get a lifetime employment commitment, but you do get security. I know people who have held these positions over a long period. (I actually don't know of a situation in which traditional tenure is offered.) 

But it isn't quite correct to say that there are no research expectations. The professors of the practice are expected to contribute to the profession and, normally, to publish and have a presence at conferences and such. But the research is more likely to be in pedagogy than theory. This is the situation in computer science at several very good universities. The people are very active professionally and may be text book authors. 

These positions exist when a good university has both a commitment to research in a field, but also to high quality undergraduate education. Holders of these positions relieve the top theoreticians of teaching lower level courses, especially to large classes, but still provide high quality instructors for the undergraduates. Of course, such people are also able to interact with the more traditional research focused faculty.