Greek: καὶ ἄλλα πρόβατα ἔχω, ἃ οὐκ ἔστιν ἐκ τῆς αὐλῆς ταύτης· κἀκεῖνα δεῖ με ἀγαγεῖν, καὶ τῆς φωνῆς μου ἀκούσουσιν, καὶ γενήσεται μία ποίμνη, εἷς ποιμήν.
My Translation: And other sheep I have, which are not of this court; I should lead them too, and they will hear my voice, and one flock will happen, one shepherd.
KJV: And other sheep I have, which are not of this fold: them also I must bring, and they shall hear my voice; and there shall be one fold, and one shepherd.
My translations are purposefully stretched and should not be viewed as more accurate than the KJV translation unless I say so in the post. I'm trying to show the range lying between the original Greek text and the English.
Update May 2013 This scripture has been removed by the Church Educational System from the Scripture Mastery list. However, it had remained within this list for over two decades and as such is still familiar to many graduates of the LDS Church's Seminary program. So I'm keeping this exploration of it online, but it is no longer applicable to CES.
Let's discuss the general non-Mormon view on this scripture first, and then we'll get to how this scripture is usually applied in the LDS Church.
This verse is in the context of a much longer discourse by Jesus about how he is the good shepherd willing to die for the sheep. Don't forget this context because it's important to how everyone else reads this verse.
In verse 10 onwards, Jesus explains what he means by the term "good shepherd." He's not a hired hand, who would run away when danger appears and would leave the sheep to be destroyed. No, he's a good shepherd, who is willing to die in defense of his sheep. And Jesus says that God knows that Jesus dies for his sheep. And Jesus has other sheep, not of this enclosure, and he plans to bring them into the enclosure and have one herd. And the Father loves Jesus because he will die for the sheep, and Jesus is fully in command of the circumstances of his death. He has the power to die, and the power to take death back.
So, with that context surrounding the verse, it shouldn't surprise you that most people, believers and non-believers alike, feel that the author is describing how Jesus's death is for both the Jews as well as the Gentiles. The rest of the context in the sermon is describing how Jesus's willingness and ability to die for his sheep is what sets him apart from a hired hand as a good shepherd.
In Matthew's Gospel (which, I cannot stress this enough, is written from a Jewish perspective for what appears to be a Jewish group of Jesus followers), Jesus tells a Syrophonecian woman that he was sent only to the "lost sheep of Israel"; Jesus also himself avoids and tells his followers to avoid Samaria during his ministry. However, in John's Gospel, one of the first declarations by Jesus about being the Messiah is made to a Samaritan woman, and Jesus does not seem to share the same concerns. Also, while Matthew was written while there was still much uncertainty about the place and role of Greek believers, John was written a few decades later when the balance between Jewish Christians and Greek Christians was beginning to swing decisively in favor of Greek Christians having the greater numbers. So from this perspective it makes sense that the author of John would choose to mention that Jesus's mission to die for his sheep also extends to more sheep than just his original Jewish followers. Certainly by the time John was written it would have been obvious that the Christian message was seeing much more success in the world outside of Judea than within it.
The unique Mormon interpretation, on the other hand, is demanded by the Book of Mormon. In the Book of Mormon narrative, a group of Israelites have been led by God to the Americas. After Jesus's death and resurrection, he descends to the civilization of these Israelites and spends a few chapters in 3 Nephi talking to them and setting up an ecclesiastical organization among them before leaving. It's the theological climax of the book, featuring John's Jesus giving quotations from Matthew's gospel, and setting up his Church according to patterns recorded by Paul. Which is totally fine if you feel that the New Testament is a coherent whole (as most Mormons do). However, if you view the New Testament as a group of disparate works that do not make any attempt for agreement with each other, the appearance and behavior of Jesus in 3 Nephi is immensely troubling. (I've written more about the differences between the four New Testament gospels here.)
Anyways, the anachronisms and New Testament jumble are not the main point here. As part of his American discourses in the Book of Mormon, Jesus reveals to the people gathered around him, in 3 Nephi 15:16-24 that they are the "lost sheep" referred to in John 10:16. From 3 Nephi 15:21-24:
21 And verily I say unto you, that ye are they of whom I said: Other sheep I have which are not of this fold; them also I must bring, and they shall hear my voice; and there shall be one fold, and one shepherd. 22 And they understood me not, for they supposed it had been the Gentiles; for they understood not that the Gentiles should be converted through their preaching. 23 And they understood me not that I said they shall hear my voice; and they understood me not that the Gentiles should not at any time hear my voice---that I should not manifest myself unto them save it were by the Holy Ghost. 24 But behold, ye have both heard my voice, and seen me; and ye are my sheep, and ye are numbered among those whom the Father hath given me.
Jesus even acknowledges that many of his followers thought he was referring to the Gentiles, but he then alludes to Matthew's declaration (Matthew 15:24) that Jesus was only sent to the lost sheep of Israel (which in the Book of Mormon allusion becomes a reference to the lost tribes of Israel, referring to the lost remnants of the northern kingdom of Israel, destroyed by the Assyrians in 722 BCE). Because Jesus wasn't allowed to preach to non-Jews, then how could the Gentiles have "heard his voice"? The Book of Mormon demands that Mormons view the phrase "and they shall hear my voice" as a literal pronouncement about people hearing the actual voice of Jesus.
But Mormons do so by basically downplaying the second phrase: "them also I should bring, there shall be one flock, one shepherd." In what way did Jesus bring the American civilizations (or, as the Book of Mormon implies, other lost pockets of Israelites worldwide) into a single flock (ποίμνη poímnā) with his Jewish followers in the fold (αὐλή aulā) of Jerusalem? The only way to keep this is to jump from a literal interpretation to a spiritual interpretation in the same sentence: they're brought into the same flock because they now all have the same gospel from Jesus's voice. So the interpretation says that in the midst of a long segment where Jesus, a Jewish itinerant rabbi, compares himself to a shepherd and his followers to sheep---all of which I think everyone agrees should be read symbolically and not literally---Jesus chooses to throw out a cryptic clue to his Judean followers about far-distant cousins living elsewhere in the world, followed by yet more symbolic language about bringing them together somehow. And the only reason for this interpretation seems to be because of a statement made in a fully separate gospel, Matthew, about Jesus not going to the Gentiles, a statement invalidated by other events in the gospel of John.
Obviously, I feel that the identification of the "other sheep" as Gentiles is completely correct, but that is because I feel that the Book of Mormon is a 19th Century creation. I do not feel that I am limited or forced in my perspective by what it has to say, and I'd even encourage believing Mormons to adopt a viewpoint of a 19th Century Book of Mormon to deal with issues like this. The narrators of the Book of Mormons apologize more than once for errors that may be in the book, so perhaps Mormons should adopt this verse in 3 Nephi as one of those errors. However, without a more flexible interpretation Mormons will continue to confuse other Christians when they point to John 10:16 as though it is biblical evidence for the Book of Mormon. It isn't.
Why Do I Think This Is Part of Scripture Mastery?
This scripture is kept because it is quoted by Jesus in 3 Nephi as biblical evidence for the existence of Book of Mormon peoples. However, this quotation introduces unnecessary complexity and really only stands as a valid interpretation after the Book of Mormon is already accepted as divine scripture. Without the interpretation provided by the Book of Mormon it is a scripture where Jesus prophesies how his gospel will go forth to the Gentiles (which, considering that the gospel of John was written nearly a century after the historical Jesus probably died and many Greeks were already believing in him as a divine figure, isn't really that impressive of a prophecy), and is a statement of how Jesus's death has meaning to both Jews and Gentiles together. John's main message is that Jesus died for the sins of the entire world, and this scripture stands fully in line with his main perspective and purpose.