In this context, it's a verb. In the phrase "the shining sun", it's an adjective. You can tell the difference based on the grammar pattern and word order.

It's a verb in this sentence. You're using the present continuous to describe what the sun is 'doing' at the moment. The present continuous is formed of the auxiliary verb to be ( in this case 'is') and the present participle of the verb ( in this case, 'shining').

You may find this useful: http://www.edufind.com/english-grammar/ing-forms/

It's an adjective, as it follows the "is." If you think "the shining sun..." it may be more clear. The verb form is "shine" as in "The sun shines"