Categories for the Working Mathematicians says
the fundamental notion of category theory is that of a monoid - a set with a binary operation of multiplication that is associative and that has a unit; a category itself can be regarded as a sort of generalized monoid. Chapters VI and VII explore this notion and its generalizations.
What does "a category itself can be regarded as a sort of generalized monoid" mean?
Thanks.
Per the passage itself, a more complete explanation will occur later in the text. That said:
The starting observation is that we can identify monoids with categories with one object. The intuitive direction is that for any category $D$ and any object $*\in Ob(D)$, the collection of morphisms $Hom(*,*)$ forms a monoid under composition. If $D$ has only a single object, this is the collection of all morphisms; thus, each one-object category gives us a monoid of morphisms.
The other direction may feel a bit slippery, but it's really no harder. Given a monoid $M$, we consider the one-object category $C_M$ with object $*$ and a morphism $f_m:*\rightarrow *$ for each $m\in M$, with the composition rule corresponding to multiplication in the monoid: $$f_m\circ f_n=f_{m\cdot n}.$$ Note that the object $*$ has absolutely no content - one of the themes of category theory is that it's the morphisms, not the objects themselves, that are actually important, and this is a great example of that.
Now what about an arbitrary category - that is, when we have multiple objects? Well, each object $*$ considered on its own has an attached monoid $Hom(*,*)$, as noted above, but we have additional structure present due to the morphisms between distinct objects. So a one-object category is a monoid (and indeed that's an exact correspondence), but an arbitrary category is somehow a bunch of monoids with some connections between them.
Along the same lines, groups correspond to one-object categories where every morphism is invertible (with group homomorphisms corresponding to functors between those), with "generalized groups" being arbitrary categories where all morphisms are invertible - these are called groupoids (warning: this clashes with earlier usage of the term).
We can also view other algebraic structures like rings as one-element categories with various properties or additional structure, but this gets more complicated.