"A" at the End of a Name in Polish

I think both. I can't for certain about Polish, but it is a Slavic language and thus has inflections. I will show it using my name and Russian, the details may differ, but the logic is the same.
Ivan is my name and it is the nominative case.
Ivana is the genitive case and usually means possession, but it is also the direct complement of the transitive verbs for animate nouns as the accusative case is equal to the genitive one for them.
Ivanu is the dative case and usually means the target of an action and thus is the indirect complement.
Ivanom is the instrumental case and usually indicates the way or the tool to do something, but for names it sounds somewhat silly, but it has its uses.
Ivane is the propositional case (i.e. it is always used with a preposition).
And as a final example: "I see Ivan" would be "Ya vizhu Ivana" so and 'a' at the end might be not only for possession.
Again, it may differ in Polish, but the general idea is the same.
Hope this helps.

OK, here we go:
1) Nominative: My name is Ivan = Menya zovut Ivan (Меня зовут Иван). It's kinda strange, but in this case, the nominative is more common than the instrumental.
2) Genitive: These are Ivan's books = Eto knigi Ivana (Это книги Ивана). In addition to "I see Ivan" from above.
3) Dative: Say this to Ivan = Skazhi eto Ivanu (Скажи это Ивану).
4) Accusative: The same as Genitive for masculine names, feminine names have another form: I see Vanya = Ya vizhu Vanyu (Я вижу Ваню). However the name itself is a male's name in Russian -- it is a familiar variant of Ivan, but it is inclined like a feminine noun.
5) Instrumental: I want to be (with) Ivan = Ya hochu bytj (s) Ivanom (Я хочу быть (с) Иваном). The idea of the prepositionless variant is "to be known as" or "to be addressed as". I couldn't think of another sentence without a preposition. The variant with a prepositional is more common for names though.
6) Prepositional: I think about Ivan = Ya dumayu ob Ivane (Я думаю об Иване).
Polish also has a vocative case that is lost in Russian (except in some set expressions). And in Russian it is the process of being replaced by a new grammar construct. By the way, the order is also according to the Russian grammar in as learned in Russia, the order of cases may differ otherwise.
Moreover, this is just a tip of an iceberg as these examples only apply to masculine names (and nouns) ending in a consonant. Feminine names ending in a vowel have another set of endings, and thankfully names not matching these two patterns are usually not inclined. Other names (last names and patronymic) are somewhat similar to these, but nouns and other parts of speech have more rules and exceptions.
Nevertheless, the grammar of any Slavic language is a rather complex thing, so if you have more questions -- ask away, but I can only answer based on my knowledge of the Russian and Bulgarian, the latter of which doesn't have cases.

Yep, there's a set rules whether last names change according to gender or not. In general simple Slavic ones do, like Ivanov -- Ivanova, while others don't. And then depending on this and on the gender of its bearer they are inclined out not. But I'm not sure if it is applicable to Polish.