What exactly is a $\delta$ neighborhood The set of all point $x$ such that $|x-a| < \delta$ is called a $\delta$ neighborhood of the point $a$. The set of all points $x$ such that $0<|x-a|<\delta$ in which $x=a$ is excluded, is called a deleted $\delta$ neighborhood of $a$ or an open ball radius $\delta$ about $a$.
I don't understand this definition (and because of that also the definition of a limit point):


*

*Is $\delta$ just a random positive integer?

*What exactly is the use of a $\delta$ neighborhood, I don't see how it could be meaningful at all.
 A: $\delta$ is usually a positive real number. Given such a number and another number $a$, one can talk about a $\delta$ neighborhood of $a$. As you definition states it, the neighborhood is simply the set of numbers that have a distance to $a$ of less than $\delta$. Since we talk about distance, we would want the $\delta$ to be positive.
To write it a bit differently, the $\delta$ neighborhood of $a$ is the set
$$
\{x\in \mathbb{R}: \lvert x - a \lvert < \delta\}.
$$
This can also be written as
$$
(a-\delta, a + \delta)
$$
(i.e. the open interval of length $2\delta$ centered at $a$).
So it really is not more than an interval.
When you talk about the "deleted" neighborhood, you are simply removing $a$ form the set. So the "deleted" $\delta$ neighborhood of $a$ is just the union of the two intervals:
$$
(a-\delta, a)\cup (a, a + \delta).
$$
As to the question about what this is good for, this comes up a lot of places. If you for example look at the definition of what a limit is, you will come across these neighborhoods. I suggest that you look of this definition and try to study it for a bit. Maybe that will help to make it more clear what these $\delta$ neighborhoods are good for.
A: The definition of a continuous function $f$ for example is that for any $\varepsilon>0$ of your choosing there is $\delta$ such that if $|x-y|<\delta$ then $|f(x)-f(y)|<\varepsilon$. Meaning that for point $y$ close enough to $x$ the function $f$ takes $x$ and $y$ to a close values.
In general $\delta$ neighborhood simply means all the values that the distance between them and $x$ is less or equal to $\delta$. In $\mathbb{R}$ you get that  $\delta$ neighborhood of $x$ is $[x-\delta,x+\delta]$ in  $\mathbb{R}^2$ you get an open ball with radius of $\delta$.
A: A look on this article should be helpful. 
To be brief we are interested in a neighborhood because sometimes we wanted to know what is the behavior of the function "locally" around a point. In the real line, a good way to make this "local" concept precise is to choose a sequence of neighborhoods that become smaller every time, and observe the behavior of the function in such neighborhoods. 
For example, intuitively a function is continuous if and only if for small enough neighborhood around $x_{0}$, the value $f$ in the neighborhood does not deviate from $f(x_{0})$ too much. Formalizing this we get the so called $\epsilon-\delta$ calculus, which is quite useful in formalizing various mathematical concepts like differentiability. 
