Why the word 'space' is used with metric space? I am new to math. According to the Wikipedia, A metric space is a set for which distances between all members of the set are defined. I have a silly question: Why they used the word 'space'? Why not 'metric set' for example? 
Many thanks for your answer in advance.
 A: The word "space" here is used when we are interested in ideas like continuity. There are other types of spaces which don't have related metrics - a metric space is essentially the simplest idea of a "space." It is the beginning of the topic of "topological spaces."
There are also some topological spaces that have more structure than a mere metric can provide - for example, the topic of differentiability requires something more than a metric to be defined on a space.
One thing you'll find out is that different metrics can sometimes give you the same notion of continuity. This is because the notion of continuity has less to do with the specific way we measure distances in the "space," and more about what we consider to be "near" each point $x$. So in the future, you'll start talking about neighborhoods of points. 
That word "neighborhood" indicates the intuition of a space - we are not just talking about a set with a function, we are talking about something where we have an intuition about "nearness."
A metric space has some features that a more general space does not. In particular, you can define the notion of uniform continuity on a metric space, but this notion cannot be defined on more general spaces.
There are other areas of math where the word "space" comes up - like vector space and measure space - where it does not always correspond to the above type of space that I'm discussing above. In particular, lots of common vector spaces used in math don't have a natural notion of nearness. 
A: Generally speaking, good terminology should appeal to our intuition. Your idea of a "metric set" has that disadvantage that the word "set" applies to just about everything in mathematics and so it gives us no intuitive hook whatsoever. 
In the case of the word "space", this is a clue that there are geometric concepts lying around and that you should be prepared to engage your geometric intuition. 
So, for example, a metric space is founded on a formalization of the geometric concept of distance, and you should be prepared to engage your intuition about distance when reasoning about metric spaces. 
Other examples in the very nice answer of @ThomasAndrews contain different appeals to our geometric intuition: in a vector space we can employ our intuition about lines, planes, parallelograms etc.; in a measure space we can engage our intuition about area.
