I read of “what kind of an umbrella” in a magazine, and has been confused about the expression, for dictionaries say “what kind of umbrella” is correct. Then, how is it about the article “an”? Thank you.

The phrase “what kind of umbrella” is more correct because it refers to the class of objects known as umbrellas.

However, that is a very common grammatical irregularity - the person in the magazine is asking what type of umbrella a particular umbrella is perhaps - it could be construed as more concise, however it probably is being used incorrectly.

Either way, neither form would probably greatly offend the grammar police.