I'm considering going back to graduate school, but I've heard from a number of friends that graduate students are all required to teach. Is that the case? I have no teaching experience at all. Will that negatively affect my chances of getting in?
In general, no, it won't. Having teaching experience might weigh in your favor in exceptional circumstances (a graduate department that needs a lot of teaching assistants, and you're "on the bubble"; you're going into an education program or something similar; or the application specifically asks for teaching experience).
However, most graduate schools don't expect that students have prior teaching experience, and provide training to smooth the transition.
Teaching load also varies widely from program to program: some science and engineering students TA for one semester over a five-year program, while humanities graduate students may have to TA every semester to pay for their studies.