Is 'an' necessary? What is the difference with it and without it?
In case of an emergency, push this button.
In case of emergency, push this button.
Both of the sentences work and mean almost exactly the same thing.
When you use an article, such as "an", you tend to limit the scope of what you are speaking of. In this case, to say "an emergency" clearly suggests there will only be one emergency at a time, one singlular event. It doesn't foretell what that event might be, but it will be one, clear event.
To leave out the article, it is more general, perhaps implying that when the need arises to push that button, you won't even be able to correctly identify (and thus limit) the bounds of the urgency.
It doesn't worth with all nouns. Some nouns are countable (only) - like cat - and others are generally non-countable - like happiness. You need articles for the countables, and don't need them for the non-countables. (I am looking for a cat; I am looking for happiness.) But some nouns work either way, and their meaning shifts only a little depending on how you decide to portray them.