Well, this thread is what finally made me register, congratulations.
I remember when I first realized this. I was young, probably not even double-digits, and it was late at night when my parents had already gone to sleep. I crawled out of bed, in tears, and woke my parents up crying "I'm going to die someday." Sorry mom, I'm sure that was a little unnerving.
But, after being tucked back in, I concluded it wasn't worth worrying about. But I've been thinking about this subject a lot lately, and I've found it still terrifies me. I don't want to die. I've played RPGs, I know immortality and agelessness is overrated, and I understand that the looming shadow of death is what keeps people productive and gives them purpose, but that's no comfort to me.
I don't want to die and have the world trudge on without me. I enjoy life, and I don't want it to end. I want to see all the things to come in the future.
But, seeing how most of the replies have been either "It's silly to worry about that" or "I try not to think about it", I get the impression this isn't something you're ever supposed to be comfortable with.
In the end, I guess the best you can do is not let the fear of death interfere with your life. I'm still working on that.