The word "neighbourhood", was introduced into mathematics in order to express the notion that some points are close, or near, to others. Just as you do not identify all the inhabitants of your country as your neighbours, so should not all points of a topological space be identified as neighbours. However, Topology is somewhat more lenient than Geography, so it allows you to declare an entire space as a neighborhood of each point.
The notion of "nearness", or "closeness", between points in a general topological space is more abstract than it is when there is a metric that induces the topology. A metric introduces the notion of distance between points, and so it is easy to say when is a point $x$ close to a point $y$. Clearly if $d(x_1,y_1)<d(x_2,y_2)$, we understand that as much as $x_2$ is near $y_2$, $x_1$ is even nearer to $y_1$.
Now forget the metric, and suppose you only have a topology on some set $X$. Pick two points in $X$, $x$ and $y$. How can you use the topology on $X$ in order to introduce a notion of "nearness" between $x$ and $y$? Well, all you have is two points and a family of subsets of $X$ satisfying a few axioms of topology. What can you do? You could ask if any of the sets in the family contain both of the points. Think about the family of sets as a family of geographical regions; if you find a region containing both your points, you would like to think of them as near to each other. Of course, the entire region always contains both of them, but that's not very helpful. Suppose you have two different regions containing both points. Then you can form even a potentially smaller region containing both of them – by intersecting both regions. So the more regions containing the points $x$, $y$, the closer they are to each other. What if both points belong to all the regions at once? Then depending upon the choice of regions, you can sometimes deduce that the points are equal. For example, if your topology has some property of separation.
The "regions" are of course the open sets of the topology. The nearness of points is expressed by the word 'neighbourhood'. That's why it is reasonable to think about neighbourhoods as open sets. That said, it should be emphasized that although the notion of being neighbours is naturally linked to the notion of being near or close to each other, the choice of the term 'open sets' is quite arbitrary, and we could just as well develop the theory of topology with 'closed sets'.