I've been told by numerous people that (1) my undergraduate university will be disinclined to bring me on as a graduate student, and that (2) it's a bad idea to attend grad school where you completed your undergraduate degree, anyway. Is that true? If so, why?
I will address the two points separately:
If you have a strong application, then your undergrad university will be more inclined to take you as a graduate student. The reason is that to have a strong application, you need very good reference letters. The reference letters are probably from professors at your undergrad school, and thus they will carry a lot of weight there (compared to other schools where your former supervisors are less known). Thus, it is often easiest to get into the school you graduated from.
I've heard the second point myself, and I actually advice it/try to follow that advice, too. The reasoning behind it as that at the school you graduated from, you already know everybody. Thus, if you stay you will continue to work with the same people and won't meet new collaborators. If you go to a new school, you will have to meet new people and expand your network.