Predestination is the doctrine that before we were created, our Omniscient and Sovereign God chose the elect for His salvation (sheep) and the damned for destruction (goats). So there are a few pieces of this.
Predestination in Scriptural Context
Romans 8:29-30 For those God foreknew he also predestined to be conformed to the image of his Son, that he might be the firstborn among many brothers and sisters. And those he predestined, he also called; those he called, he also justified; those he justified, he also glorified.
Ephesians 1:5 he predestined us for adoption to sonship through Jesus Christ, in accordance with his pleasure and will
Ephesians 1:11 In him we were also chosen, having been predestined according to the plan of him who works out everything in conformity with the purpose of his will,
Supporting Scripture
Romans 9
14 What then shall we say? Is God unjust? Not at all! 15 For he says to Moses,
“I will have mercy on whom I have mercy,
and I will have compassion on whom I have compassion.”[f]
16 It does not, therefore, depend on human desire or effort, but on God’s mercy. 17 For Scripture says to Pharaoh: “I raised you up for this very purpose, that I might display my power in you and that my name might be proclaimed in all the earth.”[g] 18 Therefore God has mercy on whom he wants to have mercy, and he hardens whom he wants to harden.
19 One of you will say to me: “Then why does God still blame us? For who is able to resist his will?” 20 But who are you, a human being, to talk back to God? “Shall what is formed say to the one who formed it, ‘Why did you make me like this?’”[h] 21 Does not the potter have the right to make out of the same lump of clay some pottery for special purposes and some for common use?
22 What if God, although choosing to show his wrath and make his power known, bore with great patience the objects of his wrath—prepared for destruction? 23 What if he did this to make the riches of his glory known to the objects of his mercy, whom he prepared in advance for glory— 24 even us, whom he also called, not only from the Jews but also from the Gentiles?
As we can see, and also logically deduce based on the Omniscient and Creator attributes of God, that God chooses to create both the saved and the damned with full knowledge of their final destination. We all have a purpose in His Creation, but we don't get to decide to save ourselves. That is for God to decide (and He has already done so even prior to the Beginning).
Scripture regarding the elect
Matthew 24:22-31New International Version (NIV)
22 “If those days had not been cut short, no one would survive, but for the sake of the elect those days will be shortened. 23 At that time if anyone says to you, ‘Look, here is the Messiah!’ or, ‘There he is!’ do not believe it. 24 For false messiahs and false prophets will appear and perform great signs and wonders to deceive, if possible, even the elect. 25 See, I have told you ahead of time.
26 “So if anyone tells you, ‘There he is, out in the wilderness,’ do not go out; or, ‘Here he is, in the inner rooms,’ do not believe it. 27 For as lightning that comes from the east is visible even in the west, so will be the coming of the Son of Man. 28 Wherever there is a carcass, there the vultures will gather.
29 “Immediately after the distress of those days
“‘the sun will be darkened,
and the moon will not give its light;
the stars will fall from the sky,
and the heavenly bodies will be shaken.’[a]
30 “Then will appear the sign of the Son of Man in heaven. And then all the peoples of the earth will mourn when they see the Son of Man coming on the clouds of heaven, with power and great glory.[c] 31 And he will send his angels with a loud trumpet call, and they will gather his elect from the four winds, from one end of the heavens to the other.
Romans 11:7-10 New International Version (NIV)
7 What then? What the people of Israel sought so earnestly they did not obtain. The elect among them did, but the others were hardened, 8 as it is written:
“God gave them a spirit of stupor,
eyes that could not see
and ears that could not hear,
to this very day.”[a]
9 And David says:
“May their table become a snare and a trap,
a stumbling block and a retribution for them.
10 May their eyes be darkened so they cannot see,
and their backs be bent forever.”
2 Timothy 2:10 Therefore I endure everything for the sake of the elect, that they too may obtain the salvation that is in Christ Jesus, with eternal glory.
Titus 1:1 Paul, a servant of God and an apostle of Jesus Christ to further the faith of God’s elect and their knowledge of the truth that leads to godliness—
1 Peter 1:1 Peter, an apostle of Jesus Christ, To God’s elect, exiles scattered throughout the provinces of Pontus, Galatia, Cappadocia, Asia and Bithynia,
2 Peter 1:10 Therefore, my brothers and sisters, make every effort to confirm your calling and election. For if you do these things, you will never stumble,