They let you set out criteria, or what counts in the work, with levels that you can select depending on how well each student meets each criterion. You can give feedback only (no numeric grade) or you can define points for each level, which automatically calculates a grade. You can leave a comment on each criterion. Rubrics have three broad functions: a standard by which to grade students' work which is also available to students; a vehicle for feedback; and evidence for the teaching team. Rubrics can make marking fast and feedback straightforward to read, and can also help students to understand the marking criteria before they write and submit their work.
Rubrics contrast with Marking Guides (or Grading Forms in Turnitin) - the latter allow you to specify a criteria but instead of levels there is only a numeric grade to give.