Examples of large categories are the category of sets, of groups, of topological spaces, of rings, of vector spaces, of modules, of ....., also, any proper class gives rise to two large categories, the discrete one and the indiscrete one on the set.
The importance of size in category theory is an issue of the existence of certain constructions. For instance, it is useful to know if a category admits certain limits or colimits, since if it does, then you know you can always construct a certain object as the (co)limit of some diagram in the category. If a category is a complete lattice, then it has all limits and colimits. All here means no restriction at all on the diagram. In the other direction, if a category has all limits or all colimits, then it actually must be a poset (and thus a complete lattice).
This shows that there is some mutual exclusivity between having all (co)limits and having more than two arrows between objects. Since we are quite often interested in categories that are not posets, and we are also interested in having (co)limits, we must set some size constraints on the diagrams for the (co)limits. Typically, one defines a category to be small (co)complete, if it has all small (co)limits. That means that any diagram indexed by a small category will have a (co)limit. Large diagrams may or may not have limits.
Also, without the restriction of size, the difference between limits and colimits becomes blurred. It is possible to exhibit any limit as a (potentially) large colimit, and vice versa. Same holds for left and right Kan extensions.
Size issues also play a role in knowing that functor categories exist and also in constructions of left/right adjoints, but I won't get into that since my answer is long enough, and I hope it answers your question sufficiently, at least for now. And see Importance of 'smallness' in a category, and functor categories.