$\delta$ 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.