My 2 cents as a native English speaker:
In my observations, zwar usually seems to fill the same role as the English indeed / in fact.
Quick note about these two terms:
Indeed and in fact are roughly synonymous in that they both pretty much amplify something in the same manner that zwar does, except have slightly different connotations. Indeed usually leans toward reinforcing something that has already been established, while in fact leans more toward introducing a new concept, possibly also contradicting an already-established concept.
Deciding which one to translate zwar to depends on the context of the conversation.
To translate some of the examples given in fellow answers & the OP:
Ja und zwar mein Bruder
Yes, my brother in fact does!
Zwar erinnerte ich mich an seinen Geburtstag, doch ich rief ihn nicht an.
I did indeed / in fact remember his birthday, though I didn't call him.
Zwar klingt das unglaubwürdig, aber es stimmt wirklich.
It indeed sounds incredulous, but it's actually correct.
Du bekommst Ärger, und zwar gehörigen.
You'll be in trouble; you will indeed.
The biggest stretch for this is when it means "namely":
Ich spreche zwei Sprachen, und zwar Deutsch und Englisch.
I speak two languages, and they're in fact German and English.
That sounds a little odd, and "namely" would certainly be preferred. But it's still not even completely wrong. It can work much better if you were already talking about the subject with someone and thus use indeed instead.
Translating is of course a complicated process with many factors, and you'll very rarely get a 1:1 "perfect" translation for any word, and I do not claim such in this answer. For most of the translations I've provided I'm sure there are "better" versions, and I do not claim that indeed / in fact is always the "best fit" when translating zwar. However, I've merely noticed that it helps to mentally map to a similar concept in your native tongue. At least for me personally, when I encounter zwar in a sentence, I start with one of these two as a launching point for my brain, and from there on it becomes much easier to deduce the meaning and, subsequently, translating to a roughly equivalent, less awkward form is a simple task.