A lot of people gave good answers. I tutor in math and sometimes in chemistry, right now I'm hired through my school but occasionally I'll tutor privately, I've been doing it for 2 years.
Like people said, expect flaky people. Personally, if you cancel within 4 hours of when I am going to tutor you, I will charge you anyway. I plan my schedule around you, so you give me the same courtesy. Also, expect people that seem to never have been to class in their life. OMG.
If you don't know the answer, just ask your student to wait while you look it up in the book (it's funny how ALL the answers are in the book, but your student can't be bothered to open it 80% of the time). If you can't find the answer in a reasonable time, just apologize and get them the answer next time, email it to them, or have them ask their teacher. Don't sweat it, unless it starts happening all of the time.
After you tutor for a while you will be able to gauge if people are smart but just lazy, smart but need confidence/a little extra help, or if they really are, err, slow. The main thing is motivating people, which usually happens on it's own after they see a rise in grades. However, some people just lack motivation and just don't care. lol. I sound blunt but that's the way it is. Some people I find myself pushing, especially if they are really behind. You will come across some people who, in your case, don't have a proper grasp of the English language, not know what a noun or verb is and be bewildered that they made it to college. To those people I try to nicely but firmly convey that they are behind for their age, really need to learn these skills and may find it hindering their ability to find a job and social skills if they do not learn these things. I try to give them a wake-up call without being too harsh. But I have seen a tutor make someone cry....lol.
Tutoring is fun, sometimes it can be trying, but I really enjoy it. If you know their parents or teacher give them progress reports (verbal) b/c it will motivate your student. It's a great job and pays well too!