It is helpful for you to understand the assignment from the onset, so ask the student questions to help clarify what is required. If available, look at the actual assignment and make sure the student understands the task. Remember your goal is to help students improve their writing, not to create perfect papers. Jeff Brooks writes in “Minimalist Tutoring: Making the Student Do All the Work,” 

The primary value of the writing center tutor to the student is as a living human body who is willing to sit patiently and help the student spend time with her paper. This alone is more than most teachers can do, and will likely do as much to improve the paper as a hurried proofreader can. (qtd. in Murphy and Sherwood 84)

Writing usually improves with each revision, and when a student comes to the Writing Center, you become the audience for the revision process. Often a student need a sounding board; he can correct his own problems  when he takes the time to concentrate on his paper. Just as a camera lens clarifies images, you can help the student see his writing in sharp relief.

As a tutor, you don't  need to know everything; however, you do need to be a good listener. Beverly Lyon Clark writes in Talking about Writing: A Guide for Tutor and Teacher Conferences:

What many students need most is a sympathetic listener, someone who can be understanding, who can listen as they talk about a paper . . .. And that’s easy enough to do--you don’t need special knowledge in order to listen . . .. Paradoxically, you’re often most helpful when you feel as though you’re doing least. (3)

Remember to use all human resources available: Writing Center Directors, faculty, etc.  Also use resources such as dictionaries,  handbooks, and a thesaurus, when necessary. If you use these tools, students will learn to use them also.