A large category is a category which is not small.
Many tools and results about small categories, in particular concerning limits indexed by such a category, fail for large categories. There are various notions and techniques to deal with this problem and reduce or relate large categories to small categories as much as possible:
In the simplest case the large category is essentially small in that it is equivalent to a small category. Then (for all non-evil purposes) you can simply replace it with a small category.
Many large categories that arise in practice are (even essentially) large but still accessible. An accessible category is a large category which behaves like the category of ind-objects of a small category and is therefore, while itself large, entirely governed by a small category.
See also: foundations, locally small category.