There is a big apple in the refrigerator. (an should be changed to a before the word "big" because "big" starts with a consonant sound)
It took me an hour to finish my homework. (the should be changed to an before "hour" because "hour" starts with a vowel sound)
I don't know the full name of my manager. (a should be removed before "full" because "full" is not a countable noun)
They are the most beautiful creatures on earth. (creature should be pluralized, and the word "the" should be added before "most" because it refers to a specific group of creatures)
Mary is studying at a university in Hanoi. (no mistake)
This is a unique custom of this ethnic group. (an should be changed to a because "unique" starts with a consonant sound)
Did you enjoy the party last night? (a should be removed because "party" is not a countable noun)
The sun is shining brightly. (no mistake)
No one in my family likes drinking beer. (a should be removed before "beer" because it is an uncountable noun)
Peter is a friend of mine. (no mistake)