"My second grandfather"
From what I found, it is "mein zweiter" and "der zweite". It has to do with declension (http://en.wikipedia.org/wiki/German_declension , look at "Attributive adjectives"). It seems that you either have a definite article, or you have to add the ending to the adjective. For example, 'der erste Mann' but 'erster Mann'; 'das kleine Kind' but 'kleines Kind'. And that is Nominativ.
I found some additional rules in a German grammer book that discusses the case where an attributive adjective (i.e. second) is followed by an "ein" word: Masc = -er(N), -en(A), -en(D), and -en (G) Fem = -e, -e, -en, -en Neut = -e, -e, -en, -en Plur = -en, -en, -en, -en *Source: Schaum's Outlines: German Grammer