Aside from necessarily being sliced, the only other rule here is that it can't be too hole-y (or your cheese will drip out), and it can't be sliced too thick (lest your cheese won't melt). White bread and American is what many of us grew up on, but if you want to go fancier, feel free to use a nice hand-sliced Italian ciabatta, a good sourdough, or a French boule. Grilled cheese is a great way to use up day-old bread, as the grilling process will resuscitate it a bit.
I use either Pepperidge Farm or Arnold white sliced sandwich bread for my grilled cheese. It's got a bit more substance than Wonderbread and its ilk, but not so much that it becomes tough. If you can get your hands on it, Japanese-style shokupan will make the finest grilled cheese you've ever had.
The Bread