Which of these is better to say?
An English exam will be on Monday.
English exam will be on Monday.
The English exam will be on Monday.
Not the middle option, but the rest of it really depends on context.
Dolores uses the indefinite article because she's referring to each exam as one generic test. She could also say:
Those both mean the same thing.
Because the person mentioned the exams in the previous sentence, the definite article is used, thus pointing back to the preceding sentence:
I'm having trouble thinking of a context where I would use no article, unless the word exam was pluralized:
The version with the is fine, assuming it's contextually appropriate—most likely, a test has already been mentioned and "the test" refers back to it. Here, "the test" is probably old information, something the speaker expects the listener to know about already.
The version with a is strange because "a test" represents new information, and in English there's a broad preference for subjects that represent old information. To avoid this, we can use an existential construction with the dummy subject there:
This sentence is better. It's not a grammatical requirement, though—the non-existential version is also grammatical. There are other ways to avoid placing new information in subject position; as in J.R's answer, you can instead write:
Here, the subject I represents old information (in most cases the listener already knows that the speaker exists).
The version without an article is ungrammatical, though in informal speech it might be elided.
You must use an article.
If you are talking about a particular English exam, use “the”. If you are talking about any English exam, use “a”. Usually, I think, you would have one particular exam in mind, so you would use “the”.
But you could also say:
Here you're not worried about the particular test, just that there is one.