Predestination is the act of God choosing who will be saved and, through His work, ultimately saving all of them. Meanwhile, none of the people He didn't choose will be saved.
This isn't a discouraging doctrine. It is a necessary doctrine, and it is a comforting doctrine when combined with everything else taught in Scripture.
So, here are the key points:
1. Sin and our inability to come to Christ by our own free will
First, we are all sinners and incapable of coming to Christ by our own free will. We're just not able to turn from sin to Christ on our own. Period. So, if left to our free will, we're all on our way to hell. See especially the red font below:
--Jeremiah 13:26, "Can an Ethiopian change his skin, or a leopard change his spots? Neither can you who do good who are accustomed to doing evil."
--John 6:44a, "No man can come to me, unless the Father which sent me draw him."
--1 Corinthians 12:3b, "no man can say that Jesus is Lord, but by the Holy Ghost."
2. If we cannot come on our own, then God has to bring us.
Second, if we cannot come to Christ by our own free will because we are sinners, then God is the one that has to bring us. This is all over the Bible, of course, but the green font in the verses quote above show this, too. God has to bring us to Christ. And when we come to Him, it wasn't because of what we did. It was because of what He did. Hence, we must be "born again." We didn't choose to be born the first time, and we didn't choose the second time, either. God is the author of life -- both our natural life and our spiritual life.
3. God does not bring everyone to Christ.
This is the hard part for many people to accept. They want to say that God is trying to save everyone, but a lot of people just won't agree to it. So, His Spirit is operating on everyone, doing His best to save them without violating their free will. If this is all God did to save us, then no one would be saved since our free will doesn't do good on its own. But here's what Scripture has to say about His passing over some people:
--John 17:9, "I pray for them: I pray not for the world, but for them which thou hast given me; for they are thine."
Here, Jesus explicitly excludes many people in his prayer. He is only praying for the people the Father has given him -- and that is not everyone.
--Matthew 11:25-27, "At that time Jesus answered and said, I thank thee, O Father, Lord of heaven and earth, because thou hast hid these things from the wise and prudent, and hast revealed them unto babes. Even so, Father: for so it seemed good in thy sight. All things are delivered unto me of my Father: and no man knoweth the Son, but the Father; neither knoweth any man the Father, save the Son, and he to whomseover the Son will reveal him."
In this passage (see the red), Jesus thanks the Father for hiding the knowledge of Christ and the truth from many people, and only revealing these things unto some others (i.e., "babes"). In other words, there is a part of the world from which God is hiding the things of salvation -- and Jesus is thanking Him for doing it this way. Notice that no man can know God unless He is one of the people to whom the Father and the Son reveal Him (see the green), and this passage explicitly says that this isn't everyone. So God is not bringing everyone to Christ.
This means that God chooses some people to be saved, and He excludes others from this gift.
Now, the question is, why is this just?
Well, we did have free will at one point -- in the Garden of Eden. But we sinned in Adam. So, now, humans are sinners by nature and deserve to be condemned by God. God does not owe us a way of salvation any more than God owes fallen angels a way of salvation. (And it is interesting to note that God does not offer a way of salvation to fallen angels, but nobody complains about that -- probably because we're not fallen angels.)
In other words, since God doesn't owe any of us mercy, He doesn't have to save everyone. In fact, He doesn't have to save anyone. And the real mystery is that He saves anyone at all.
Finally, Christ's atonement is sufficient to save anyone who comes to him -- even if every person on earth converted. And God promises salvation to anyone who comes to Christ. So this doctrine should not be a source of discouragement to anyone. We can honestly call everyone to Christ, and God will -- He promises -- save any who come to Him. It is confusion about the doctrine that causes problems at this point. The doctrine itself is consistent with a universal offer of the Gospel.