In LLPSI, I have seen Latin be Latina, Latinum, and Latinae. What are the differences of these words and how do I determine which to use?
That will depend on a number of different factors. Latin is a heavily inflected language; we do this in English (think he, his, him, etc), but not to the degree that Latin does. Like in English, the ending is telling you about the word and how it's being used in a sentence.
Let's look at Latina in LLPSI. The -a ending tells you that it's an adjective that applies to Lingua. As an adjective the ending will match both gender and number of the noun that it describes. If you use "latin" to describe a masculine or neuter word, then it would use a different ending. If it was describing something that was plural instead of singular, then it would have a different ending still (with the gender female, latina would become latinae as an example).