It is describing elephas, but through the verb est. This is called a predicate adjective. In other words:
So magnus is not an adverb and not part of the verb itself; it is an adjective linked to the subject by est.
Est is the third-person singular present tense of esse, meaning to be. So it means is.
Because the subject elephas is singular, Latin uses est, not a plural form like sunt.
Latin word order is much more flexible than English word order because the endings of words show their roles in the sentence. Putting est at the end is very common and sounds natural in Latin.
So all of these are possible Latin word orders:
They all mean roughly the same thing, though the emphasis can shift a little.
In a common classroom pronunciation of Classical Latin, you could say it approximately like:
EH-leh-phahs MAHG-noos est
A few useful points:
Different Latin pronunciation traditions exist, but that approximation is a good start for a beginner.