Clinical Law Faculty usually supervise one or two law clinics where law students can get practical experience getting indigent or immigrant clients. In this case, "clinical" means that their responsibilities are mainly focused in the teaching and running of said clinics and that they are not full faculty in the law school in terms of voting and tenure rights.
Clinical Writing Faculty is a term I've recently come across for some professional staff (with PhDs) who work in university writing/tutoring clinics. They help students with their essays and may also teach some composition courses. These people are also not full faculty in terms of voting and tenure rights.
Note: "Adjunct" is not appropriate for these faculty as they are often full time with renewing terms. "Visiting" is inappropriate as they are on renewing terms that could span decades of service. "Staff" is inappropriate as they have terminal degrees and are teaching/supervising students. "Professor of Practice" is a mostly equivalent term to "clinical faculty," again with the emphasis on teaching rather than research.