The correct pronunciation is AR-a-gorn. In the Sindarin langauge, according to pronunciation keys in the appendices to The Silmarillion and LOTR, vowels are pronounced long. A would be Ah, as in "father," "Bach," etc.
According to how Elvish are pronounced, it's Ah-rah-gorn. Not Eragorn. The latter is a result of anglicization of the name. But Aragorn is not English, nor American, it's Sindarin.
Aragorn is sometimes pronounced Er-a-gorn in the Lord of the Rings movies, but I have noticed that Legolas and occasionally others sometimes call him Ar-a-gorn.