Bright is an adjective and brightly is an adverb.
- The bright lights hurt my eyes. (Here the adjective bright modifies the noun lights.)
- The sun was shining brightly. (Here the adverb brightly modifies the verb shines.)
Sometimes the word bright is also used as an adverb.
Thus the sentence ‘The sun was shining bright’ is as correct as the sentence ‘The sun was shining brightly.’