Hello,
An honors degree
The indefinite article 'a' become 'an' when preceding nouns (words) starting with the vowels ( u ,a ,e,o,i ) and also when preceding nouns starting with (h) ,where the 'h' is not really pronounced as in the word 'honor'. It is rather pronounced 'Onor', so it starts with an 'o' sound.
Notice:
an honor
but
a house ( since the 'h' as a consonant is pronounced in this word ).