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The two translations are Nominative: "mein zweiter" and Accusative: "meinen zweiten". Dative will be "meinem zweiten". I hope a native speaker can confirm this and also explain why the first two are acceptable translations and the third is not.
The dative would mean something like "to my second grandfather", while the first two are subject and direct object, so they mean "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.
See the discussion here for jess1camar1e's awesome way to remember the adjective endings : http://www.duolingo.com/comment/556140
If you mistake "mein" for a definite article, then by the rule of weak declension, the ending for the adjective would be -e for a masculine noun in the nominative. E.g. "Der zweite Vater". This was my mistake just now.
Nom and acc are the examples given, as they both mean 'my second grandfather' in English. Dative is not, as it would mean something like 'to my second grandfather'.
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
