Does Jesus refer to aliens here? Is this a prediction that eventually we will take the Gospel to other planets?
It is unlikely that extraterrestrials as we generally imagine them are specifically referred to here. Indeed, in at least one other place when God defines who his sheep are, he specifically states they are men (Ezekiel 34:31; ESV quoted):
And you are my sheep, human sheep of my pasture, and I am your God, declares the Lord GOD.
The Bible tells us little of other inhabited worlds. We can infer that any other inhabited planets would be unfallen: Satan and his angels were expelled from the heavens to Earth (Revelation 12:7-9) and the host of the heavens is still said to worship God and declare his glory (Nehemiah 9:6, Psalms 19:1, etc.). If this is the case, taking the Gospel to them would quite literally be preaching to the choir.
Who are these other sheep?
Theories vary; a well-known (if not widely-accepted) one is the idea espoused by some Mormons such as James E. Talmadge (Jesus the Christ, p. 419):
The "other sheep" here referred to constituted the separated flock or remnant of the house of Joseph, who, six centuries prior to the birth of Christ, had miraculously detached from the Jewish fold in Palestine, and had been taken beyond the great deep to the American continent.
In more mainstream Christianity it seems most often considered that the division between the "sheep of this fold" and "sheep not of this fold" refers to the division between Jews and Gentiles. But when Jesus was approached by a Gentile from Canaan (Matthew 15:22-24), he states otherwise:
And behold, a Canaanite woman from that region came out and was crying, "Have mercy on me, O Lord, Son of David; my daughter is severely oppressed by a demon." But he did not answer her a word. And his disciples came and begged him, saying, "Send her away, for she is crying out after us." He answered, "I was sent only to the lost sheep of the house of Israel."
If we combine this text with the original, that suggests that the "sheep of this fold" he was speaking of was the lost sheep of the house of Israel, and the "sheep not of this fold" would then be the sheep who were not lost.
This idea is supported by John 10:16 itself, where Jesus states that the "sheep not of this fold" will listen to his voice, suggesting they are still obedient to him, which would not at the time apply to the Gentiles. Indeed, when Jesus was accosted for consorting with sinful people, he told a parable likening himself to a shepherd following sheep who had strayed from the main herd (Luke 15:1-6):
Now the tax collectors and sinners were all drawing near to hear him. And the Pharisees and the scribes grumbled, saying, "This man receives sinners and eats with them."
So he told them this parable: "What man of you, having a hundred sheep, if he has lost one of them, does not leave the ninety-nine in the open country, and go after the one that is lost, until he finds it? And when he has found it, he lays it on his shoulders, rejoicing. And when he comes home, he calls together his friends and his neighbors, saying to them, 'Rejoice with me, for I have found my sheep that was lost.'