Can anyone say what type of clause this is — noun, adjective or adverbial?
I am glad that you have passed the test.
Some people say that it is a noun clause. But I am not sure. What is the syntactic relation between glad and the that clause?
The clause following 'glad' is a type of subordinate clause known as a that-complement clause. Adjectives such as glad can take a complement clause modifier with 'that' preceding the complement clause and functioning as complementizer, ie to introduce the complement clause.
This complement clause cannot be defined as any of noun, adjective or adverbial as 'you have passed the test' is a complete clause and would be able to stand alone as a main clause.
ADDITION: since first writing this I have become aware that there is a tradition in ESL teaching of referring to complement clauses as 'noun clauses', so this would be a better fit to the question. However this is not standard within the study of English linguistics.
What you heard is right: it is a noun clause.
More precisely in your example, it's a noun clause acting as an adjective complement.
Your adjective is "glad" and your adjective complement is "that you have passed the test."
You can read more about it here.