In LOTR we hear a tale of two brothers, Elros and Elrond, who are given the choice of long life or eternal life. Elros chooses long life and from him descended the Numenourians. And Elrond chose eternal life and from him, the Grey Elves descended. Most Ringers know this story, what I don't know is if the Elrond who was Elros' brother was the same Elrond who accompanied Isildur to Mount Doom and formed the Fellowship of the Ring.
Yes, they are the same person. As a general rule, because elves in the Lord of the Rings are immortal, their names are very significant to them. As far as I know, there are no two elves in the novels, or any supplemental material, that have the same name. (There is one instance where Tolkien re-used the name of an elf that had previously died, but this is explained as being the same elf, reincarnated into a new body.)
So, there is only one Elrond, and he is a very important figure in the mythology of Lord of the Rings. He founded and ruled Rivendell, and fought against Sauron several times, including the Last Alliance. He aided Bilbo by translating an ancient map that used very old elvish magic. And, he was still ruling Rivendell when Frodo came there, and he's the one that formed the Fellowship of the Ring.
Although Elrond chose eternal life for himself, the choice given to him and his brother was passed down to his descendants; this is why his daughter Arwen was allowed to become mortal, which she did when she married Aragorn.