Sentence fragment: I am currently taking freshman English. A required course.
Corrected sentence: I am currently taking freshman English, which is a required course.
Therefore, the corrected sentence to eliminate the fragment results in an entire thought. It is essential in writing and speaking to convey a complete thought, especially in formal writing and speaking, such as is required in academics and in business.
Therefore, in the above, the intent of what the speaker wants to write or say is not broken into two sentences, with one sentence being an incomplete thought. The incomplete thought above is the sentence “A required course.” The corrected sentence is cohesive and gets the speaker's point across clearly.
Of course, we often speak in fragments during our typical discourse throughout the day. This is accepted and is normal. It's when we have more official communications so-to-speak to engage in that we have to be careful about sentence fragments.
We convey our dexterity with the English language when we construct proper sentences. This is essential in job interviews, in test writing, in presentations, speeches, and such. Proper English communication can be the clincher in securing a better position (career-wise) in many instances.
The issue with the way these "sentences" are currently written is that the second part, "A required course", is not a sentence because it does not have a predicate (verb). Because "A required course" describes "English" from the previous sentence, one way to combine these sentences is as the previous response states: I am currently taking freshman English, which is a required course.
A shorter way to correct the fragment error is to leave it as an appositive, a noun phrase that describes a neighboring noun. I have attached a link where you can read more about this. In this case, the sentence becomes: I am currently taking freshman English, a required course.
On the other hand, you also have the option of turning the fragment and combining it that way. One method of combining two sentences that is applicable to this case is with a semicolon. We could then rewrite the previous statements as: I am currently taking freshman English; it is a required course.