What is your definition for neighborhood in topology? As you know, Munkres-Topology and Rudin-Analysis are really widely using textbooks for undergraduates. They all define a 'neighborhood of $x$' as an open set containing $x$, so i have followed this definition for 6 months. However, surprisingly, Wikipedia defines a 'neighborhood of $x$' as a set containing an open set containing $x$.
This really makes me annoyed, since this means that whenever I find a definition referring to a neighborhood on Wikipedia, I have to check whether that definition is equivalent to my definition of a neighborhood.
Which one is widely used?
 A: From mathworld@wolfram: 

In a topological space, an open neighborhood of a point is an open
  set containing it. A set containing an open neighborhood is simply
  called a neighborhood.

In most cases proofs involve open neighborhoods, so it usually shouldn't make too much of a difference, but It does look like different text books define this differently.
A: Since the question asks "What is your definition?" I'll say that my book Topology and Groupoids (first edition, "Elements of Modern Topology" (1968)) uses the definition that a neighbourhood $N$ of $x$ is such that $x$ is contained in the interior of $N$. Thus in the real line, $[0,1]$ is a neighbourhood of all of its points except $0,1$. 
In practical terms, the difference between the two definitions is marginal, except that the neighbourhood axioms seem simpler with  the more general definition. 
I still hold to the idea that for a beginner, the definition of a topology in terms of neighbourhoods is the most intuitive and easily motivated; thus for continuity it is related to $\varepsilon-\delta$ methods in analysis. Of course students have to become familiar with the open set definition as well, including that for continuity, but should not have the idea imposed  that there is only one route to the useful concepts. 
A: Rudin's Real and complex analysis, third edition (1987), page 9 of the French translation (1998):

Définissons d'abord un voisinage d'un point $x$ comme un ensemble contenant un ouvert contenant le point $x$. (Let us first define a neighborhood of a point $x$ as a set containing an open set containing the point $x$.)

It appears (thanks to @Martin for this) that the English and the French versions disagree since, on page 9 of the third English edition there is a parenthetical remark defining neighborhoods:

(A neighborhood of a point x is, by definition, an open set which contains x.)

This decision of the French translator of Rudin's book to modify this definition backfires on him, later on in the book, on page 35-36 Definition 2.3(d): there, the English text defines again a neighborhood as open and mentions parenthetically that some authors use the other definition; and all of this is translated faithfully in the French edition, in contradiction with the choice made earlier on to modify Rudin's text. Traduttore, traditore...
Munkres's Topology, second edition (2000), indeed stipulates that every neigborhood is open and, immediately after the definition, signals the alternative definition (pages 96-97).
All in all, it seems that readers of Rudin's and Munkres's books might not be completely taken aback by Wikipedia's version since both these authors, while following the other convention, explicitly mention this one.
A: Personally, I prefer the definition of it being a set that contains an open set that contains the element. When people say that a neighbourhood of $a \in X$ is just an open set that contains the element a, it takes off some cases. I'll explain it better:
We say that a is an interior point of $X \subset \mathbb{R}$ when there is a number $\epsilon>0$ such that the open interval $(a-\epsilon,a+\epsilon)$ is a subset of $X$.
The set of the interior points of $X$ is called the interior of the set $X$, represented by $int\space X$. 
When $a\in int\space X$, we say that $X$ is a neighbourhood of the point a. 
A set $A \subset \mathbb{R}$ is called an open set when $A = int \space A$, that is, when all the points in A are interior to A. 
Being so, when a 'neighbourhood of $x$' is defined as being an open set containing $x$, it is not considering the cases in which the set in which $x$ belongs is a closed set. 
Let's say that we have $c<x<d$ and $x \in A = [c,d]$. Then the interior of $A$ is the open set $int \space A = (c,d)$. Also, we have that $x \in int \space A$, and so $A$ is a neighbourhood of the point $x$ - it is a set containing an open set that contains the element $x$. 
In the spectrum of topological spaces, one talks about open and closed neighbourhoods - in that case, the definition of an 'open neighborhood' can then be stated as an open set containing the element. 
