'eoyo' (-어요) is the standard polite, informal ending one attaches to a verb stem and 'haeyo (해요) is just an example of its usage. Basically, 'haeyo' (해요) is a contraction of 'ha-yeoyo' (하여요), where 'ha' (하) is the stem of the verb 'to do' (the dictionary form is 'hada' (하다)) and 'yeoyo' (-여요) is the irregular form of 'eoyo' (-어요) that combines with it. Note also that 'ayo' (-아요) is used in place of 'eoyo' (-어요) when preceded by 'light' vowels (i.e., 'a' and 'o') due to vowel harmony in Korean phonology. There are also a bunch of other rules for dropping vowels, consonant changes, etc. but will skip over that here.