I just started reading Categories and Sheaves and noticed that the book's definition of a category requires the collection of objects of a category to be a set. However, from other experiences, I have heard that the object set of a category can also be a class. So in the former, the category of sets doesn't contain all sets (but instead some collection of sets that live in a universe set) whereas in the latter, the category contains all sets. Presumably by only requiring the object collection to be a class, we can apply category theory to a larger collection of mathematical structures. What is a possible reason for this seeming discrepancy in definitions? By following the book's definition, will I be missing out on anything?
The book you mentioned uses, as its underlying set theory, a version of Zermelo-Fraenkel set theory with a universe. (I don't have the book, and the most relevant page, namely page 11, is omitted from the Google preview at your link, but the bottom of page 10 indicates that page 11 will introduce universes.) The point of universes is to allow you to treat proper-class-sized things as sets, and therefore to have even larger sets available. More precisely, given a universe $U$, we have the following "translation" from the usual language of class-set theory (like von Neumann- Bernays-Goedel set theory or Morse-Kelley set theory --- with sets plus proper classes, where the latter can't be elements of anything) to the language of a set theory with a universe $U$. The sets of a class-set theory correspond to the elements of $U$. The classes of a class-set theory correspond to the subsets of $U$. (There is nothing in class-set theory to correspond to even higher sets, like families of subsets of $U$.) I believe that, if you use this translation, you'll find that the class-set presentations you've encountered earlier match up with the universe-style presentations in the book you cite.