@EmpressPixie and @girlofscience
I also did the whole grad school app thing last year. After getting done with my applications, I was sure that I would get all rejection letters. Fortunately, it was the exact opposite. I second GoS, if you need any help, I’m happy to do so.
In answer to the question, rejection is the biggest fear, but if you’re asking more about the actual process, I remember hating having to write a personal statement. Scores and grades are fixed (you can’t do anything about them once you’ve got them), but personal statements are so subjective, you never know what to write.