What is a rubric?
A rubric is a document that establishes performance expectations for an assignment.
Rubrics specify several criteria by which a student's performance on a task will be judged, and they clearly define a continuum of performance levels for each of these criteria—from excellent to poor.
Scores can be attached to each of these performance levels. For example, a scoring rubric for an essay may have 5 performance levels—a score of 1 would be "poor," and a score of 5 would be "excellent." Scoring rubrics can help you show your students what writing an "A" paper would entail, what writing a "B" paper would entail, and so on.
How can rubrics help me?
It's likely that performances in all the criteria of a rubric will not be uniform; for example, a student may organize her essay very well, but her grasp of grammar may not be so great. Scoring rubrics help you take all of this into account in order to give the student an overall score for the piece of work they've produced.
Scoring rubrics also allow you flexibility. You might decide that a student's organizational skills are more important than their use of punctuation (or vice-versa). In this case, you'd weight organization more heavily in your evaluation of student performance so that organization counts for more of a student's grade than punctuation. You're constructing the rubric, so these choices are up to you.
Rubrics can be tough to construct, but they certainly make grading easier and help you consistently evaluate student work.