What are automorphisms and subgroups? EDIT: I am a complete math newbie with only calculus and linear algebra under my belt. One of the reasons I chose that book - as it's aimed at general audience.
Reading Awodey's Category Theory he says, on p. 12 (NOTE: I provide the definition for the purposes of the context only. I have no questions about that):

Definition 1.4. A group $G$ is a monoid with an inverse $g^{-1}$ for every element $g$. Thus, $G$ is a category with one object, in which every arrow is an isomorphism.
For any set $X$, we have a group $\operatorname{Aut}(X)$ of automorphisms (or "permutations") of $X$, that is, isomorphisms $f:X\to X$.
  A group of permutations is a subgroup  $G\subseteq \operatorname{Aut}(X)$ for some set $X$, that is, a group of (some) automorphisms of $X$. Thus, the set $G$ must satisfy the following...



*

*What are these "permutations/automorphisms"? He never introduced them before. 

*And what is a "subgroup"? Is it used informally here?

*I guess more general question, is what is the author trying to teach me here - the fact the group can have subgroups...?


Any help appreciated, thanks.

He introduced the permutations in that definition: a permutation of $X$ is a bijection $X\to X$. A subgroup $H$ of $G$ is just a subset $H\subset G$ that is also a group with the same operations.
What the authors might be trying to teach you is that there are a lot of groups. You will eventually learn, also, that any finite group is a subgroup of some group of permutations like those in that definition. 
