Learn German: Der Elefant ist bei weitem das größte Tier hier. - The elephant is by far the biggest animal here.

bei weitem translates to "by far" and is used to strongly intensify the superlative das größte. You can also use it with comparatives, such as bei weitem größer (bigger by far).

Can I use the other words from this lesson, like weitaus or wesentlich, instead of bei weitem in this sentence?

You can absolutely use weitaus here: Der Elefant ist weitaus das größte Tier hier. However, wesentlich and bedeutend (which mean "significantly") are generally only used to intensify comparatives (like wesentlich größer), not superlatives.

Why is it das größte instead of am größten?

Because the adjective is directly modifying the noun Tier. When a superlative comes right before a noun and follows a definite article (das), it takes a weak adjective ending, which is -e for the neuter nominative. You would use am größten if the adjective stood alone at the end of the sentence without the noun: Der Elefant ist bei weitem am größten.

Could bei weitem go at the end of the sentence?

No, in German, degree adverbs must sit directly in front of the adjective or phrase they are intensifying. Putting it at the end, like you sometimes can in English, sounds unnatural in German.