Learn Turkish: İkinci kişi için gümüş madalya var. - There is a silver medal for the second person.

The suffix for ordinal numbers (like first, second, third) is -(i)nci. Because the number iki (two) already ends in a vowel, we drop the vowel from the suffix and just add -nci to make ikinci (second).

Does the word kişi need a special case ending because it comes before için?

No, it stays in its plain form. The postposition için (for) takes the plain dictionary form when used with regular nouns like kişi (person). You only need a genitive case ending when using için with personal pronouns, such as benim için (for me).

Gümüş (silver) and madalya (medal) are both nouns. Why isn't this a noun compound with a suffix, like gümüş madalyası?

When a noun describes the physical material something is made of, it acts just like an adjective. You simply put the material before the object without adding any special compound suffixes to the second word. So it is just gümüş madalya.

Why is var placed at the very end of this sentence?

In Turkish, var (there is / there are) acts as the predicate (the core state or action of the sentence). Because Turkish naturally follows a Subject-Object-Verb word order, this predicate goes at the very end.