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.
I've heard from a number of friends that graduate students are all required to teach.
This is usually a requirement for doctoral students, and only because most doctoral students are having their education paid for by the university or grants. If you are terrified about teaching, my advice would be to visit your local Toastmasters chapter. While there is far more to teaching than being able to speak in front of others, the fear of public speaking is the largest (in the sense of provoking fear) hurdle that I've heard folks mention.
For doctoral students, you may very well be expected to teach - rare is the department with sufficient faculty and graduate student funding that using TAs is unnecessary. That being said, learning to teach in those institutions is considered part of your training - not something you're expected to have in advance.
I had a small shred of teaching experience before entering grad school, but many of my fellow students did not, and it mattered not even a little bit.