What is the difference between open ball and neighborhood in real analysis? I'm learning real analysis. 

Open ball: The collection of points $x \in X$ satisfying $|x - x_{0}| < r$ is called the open ball of radius $r$ centered at $x_{0}$
Neighborhood: A neighborhood of  $x_{0} \in X$ is an open ball of
  radius r > 0 in $X$ that is centered at $x_{0}$

I'm using Real and Complex Analysis written by Christopher Apelian and Steve Surace.
In my mind, open ball = a collection of points satisfy certain requirement = neighborhood. I do not find out any differences between open ball and neighborhood. Could any one explain it? Thanks!
 A: An open ball about x is a ball about a point $x$. A neighborhood of $x$ is commonly defined as an  set containing $x$ in its interior.
A neighborhood need not be  a ball/other trivial shape just a set containing $x$. 
A: Let's discuss the 1-dimensional case. An open ball in $\Bbb R$ is a set given by
$$
  B(x,r):=\{y\in \Bbb R:|y-x|<r\}.
$$
These sets are very important as they allow us to define the topology on $\Bbb R$, i.e. it allows us to say which sets are open and which are not. Topology is one of the key structures to work with uncountable spaces, $\Bbb R$ in particular.
The neighborhood of a point $x\in \Bbb R$ is any subset $N_x\subseteq \Bbb R$  which contains some ball $B(x,r)$ around the point $x$. Note that in general one does not ask neighborhood to be open sets, but it depends on the author of a textbook you have in hands. 
For example, if $x = 1$ then $(0.5,1.5)$ is a ball (of a radius $0.5$) around $x$, and $(0.5,1.5)\cup [2,4]$ is a neighborhood of $x$ which is not a ball (neither around $x$, nor around any other point). 
As Brian has mentioned, indeed in your case these definitions are equivalent - but this is an unusual way to define neighborhoods.
