While in everyday life, you might think of a "set" of objects being a complete group of objects that belong together, like a "set of dishes". You might also think of a "category" as being a broad description of types of objects - for example "plates" is a different category than "bowls", but they both belong to the category of "dishes".
In mathematics, these two terms take on special meanings. In this context, we call a "set" a collection of objects, just like in day-to-day life. However, unlike day-to-day life, a "set" can be defined either by a list of the objects belonging to it, or by a rule that lets us determine which objects belong to it.
Sets have a lot of special rules and axioms that they must follow, in order to have an internally-consistent system. I won't go into all of those right now, but an important one is that a set cannot have a copy of itself as one of its objects. This leads to some rules that we think might define a set as not being valid.
This is where categories come in. In the context of mathematics, categories are like sets, but have fewer rules. Every set is a category, but not necessarily the other way around. A category can have a copy of itself as one of its objects, but this freedom means that categories aren't guaranteed to have all of the nice properties that they would if they were all sets, since they don't have all of the same restrictions on them.
*Please note that this given as an example of the English usage, and I may be misremembering some of the mathematical subtlety here.