Learn Polish: Ta gra ma dwadzieścia cztery poziomy. - This game has twenty-four levels.

In Polish, compound numbers ending in 2, 3, or 4 (like 22, 23, and 24) take the standard plural form, which is poziomy. You only use the genitive plural (poziomów) for numbers ending in 5 through 9, or 0 (like 25 or 30).

Since we are learning ordinal numbers, how would I say "the twenty-fourth level" of this game?

You would say dwudziesty czwarty poziom. Notice that in Polish, both parts of the number become ordinals (dwudziesty = twentieth, czwarty = fourth), whereas in English only the last word changes (twenty-fourth).

What case is dwadzieścia cztery poziomy in this sentence?

It is in the accusative case because it is the direct object of the verb ma (has). For inanimate masculine nouns like poziom, the accusative plural looks exactly the same as the nominative plural (poziomy).

I sometimes mix up ta, ten, and to. Why is it Ta gra here?

Gra (game) is a feminine noun, which you can usually tell because it ends in -a. Therefore, it must take the feminine word for "this", which is ta. You would use ten for masculine nouns and to for neuter nouns.