Anil met . . . . . . . . American and . . . . . . . . European man yesterday.

Here, the sentence talks about the 'American' and 'European' as general classes of people. Since we are referring to general things, we need to use the indefinite article 'a/an'. 'An' is used before vowel sound article. 'European' sounds like 'yoo' and therefore 'a' article will be used.