The adjoint of a matrix has a basic property when employed in a matrix-vector product as part of an inner product. At this point, you could even use the following result as a motivation for the definition of an adjoint.
Suppose that $A$ is an $m\times n$ matrix and $\vect{x}\in\complex{n}$, $\vect{y}\in\complex{m}$. Then $\innerproduct{A\vect{x}}{\vect{y}}=\innerproduct{\vect{x}}{\adjoint{A}\vect{y}}$.
Sometimes a matrix is equal to its adjoint (Definition A), and these matrices have interesting properties. One of the most common situations where this occurs is when a matrix has only real number entries. Then we are simply talking about symmetric matrices (Definition SYM), so you can view this as a generalization of a symmetric matrix.
The square matrix $A$ is Hermitian (or self-adjoint) if $A=\adjoint{A}$.
Again, the set of real matrices that are Hermitian is exactly the set of symmetric matrices. In Section PEE we will uncover some amazing properties of Hermitian matrices, so when you get there, run back here to remind yourself of this definition. Further properties will also appear in Section OD. Right now we prove a fundamental result about Hermitian matrices, matrix vector products and inner products. As a characterization, this could be employed as a definition of a Hermitian matrix and some authors take this approach.
Suppose that $A$ is a square matrix of size $n$. Then $A$ is Hermitian if and only if $\innerproduct{A\vect{x}}{\vect{y}}=\innerproduct{\vect{x}}{A\vect{y}}$ for all $\vect{x},\,\vect{y}\in\complex{n}$.
So, informally, Hermitian matrices are those that can be tossed around from one side of an inner product to the other with reckless abandon. We will see later what this buys us.