Attending graduate school at my undergraduate university 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?
Pros


If you stay at the same school (this applies even more when you join grad school immediately out of your undergrads), it'll be a matter of remaining in your comfort zone - same department, a faculty who know you, even the same apartment/neighborhood! This can be a major factor, depending on the person concerned - the pros of staying at your Alma mater are all about convenience IMO.
If your UG department has an influential professor with whom you've worked before and are planning to continue as well, that can be very advantageous -  as having such a faculty get to know a student's work as an undergraduate can lead to a very strong recommendation (since he has accepted you in the grad program, it is reasonable to assume your work had impressed him during your undergrads).


Cons

An important advantage of going to another school is that you will be exposed to a completely different department, with faculty who may have diverse research ideas for you to work on. The department, in turn, will benefit as well as a new student from another school will cross-pollinate their department with fresh ideas. This is so important that some top universities have a strong bias against accepting their own undergraduate students into their graduate programs.