I saw . . . . . . . . old man yesterday.

Articles are used before nouns or noun equivalents and are a type of adjective. The definite article (the) is used before a noun to indicate that the identity of the noun is known to the reader or the noun is specific. The indefinite article (a, an) is used before a noun that is general or when its identity is not known. 'The' is used with both singular and plural noun while 'a/an' is used with singular noun.
In the sentence 'old man' is general noun. If the word (old) has vowel sound 'an' article is used.