The Term Neighborhood in Mathematics 
A neighborhood of some element $c$ of $\mathbb{R}$ is a set containing an open interval $(a,b)$ such that $a<c<b$.

Is $c$ here a fixed point? For example suppose an open interval $(1,4)$ such that $x \in (1,4)$. This says that $1<x<4$ or every real number between $(1,4)$ is a member of this set and can be the value of $x$. Does $a<c<b$ also mean the same thing that every real number between $a$ and $b$ can be the value assigned to $c$ or is $c$ some fixed number which is inside the set $(a,b)$? Another question is that, is it necessary that the neighborhood of $c$ be an open interval? Can it be not a closed interval?   
 A: I think that your confusion can be traced to a simple grammatical error: The definition should read:  

A neighborhood of some element $c$ is an open interval $(a,b)$ such that $a<c<b$. 

Neighborhoods aren't unique - typically each point has infinitely many of them!  So we don't talk about the neighborhood of a point, we just say that some interval is a neighborhood. 

One additional comment though: Usually, a neighborhood of $x$ is any set $N$ which contains an open interval $I$ which in turn contains $x$.   So for example, $[0,2]$ is a set which contains the open interval $(0,2)$ which as an element the point $x=1$, so $[0,2]$ is a neighborhood of $x=1$.  
In this case, $[0,2]$ is a neighborhood of every point in $(0,2)$.  As M. Turgeon noted in his/her answer, the set of all points for which a set is a neighborhood is called the interior of that set.  Thus, the interior of $[0,2]$ is the set $(0,2)$. 
A: When we write $(a,b)$ is a neighbourhood for $c\in\mathbb R$, it is indeed implied that $c$ is fixed. On the other hand, it is correct to say that for any $c\in(a,b)$, the interval $(a,b)$ is an open neighbourhood for $c$. 
Finally, neighbourhoods are usually defined to be open sets. But sometimes it is necessary to be more general, and therefore another definition for a neighbourhood  of $x$ would be a set $U$ such that $x$ is contained in the interior of $U$. In other words, a neighbourhood should contain an open neighbourhood of $x$.
A: This is as much a grammatical or logical fact as it is a mathematical one. When we say something like

The neighbourhood of some element $c$...

We are implicitely talking about a fixed element, whatever that may be, which we denote by $c$. Consequently, for the purposes of whatever discussion follows, $c$ is that fixed object.
