Everybody sins from time to time
Yes. to some extent at least. OK. Let's assume so. (Note: Shia's consider the infallible people whom they consider so are free from sin, so let's talk about people who are not those).
When we discover that we have sinned
Yeah, you mean realize that you did something wrong, yes then,
is there any way to make up for having committed that sin?
Of Course Yes. Verily, God is Oft-Forgiving, Most Merciful. Islam is nothing but submitting to God/ way of God. It is the basic definition. All you need to do is just follow that perfectly and probably better than before.
how do we make up for having committed that sin?
OK. I would love to elaborate more about the definition.
Submitting to God, means remember him as much as you can in the best way you can, in the best way which the Prophet (pbuh) recommended one can.
Submitting oneself in the way of God, is nothing but doing good deeds. By good deeds, I mean, love your brother as your own. If you're behaving with a person in some way, just imagine yourself in that place. Would it be right the way you're behaving with that person if that person were you? If yes, then it is a good deed. Otherwise, it is a bad deed. Simple. Well.... we can't trust ourselves always in deciding right, so it is better/recommended that we refer the manual as prescribed by God and the Prophet (pbuh).
The above are basic things repeated in the entire Qur'an so many times.
Verily, those who believe and do righteous good deeds, they are the best of creatures. (Qur'an 98:7)
So, your motto should be to try to become one of the best of the creatures in front of God.
What is the final result from this repentence?
Hmmm.... This thing, I can only answer in maybes... There are different ways within those above 2 general rules and methods which are highly preferred (which are preferentially prescribed by God and Prophet (pbuh) e.g Like the one others have pointed). But remember, what ever you do, if you want to expect forgiveness from God, conditions are that you strictly try to become one of the best creatures, and by that I mean, do not repeat or perform any of the deeds that could make you a bad creature.
Is it as if the sin never occurred?
No, I think our records will be preserved perfectly, unbiased.
Is it turned into a good thing?
Maybe Yes. But *conditions apply.
Is it simply that we are forgiven?
Yes, but according to God's wish.
May the creator guide us all.
In case if you want to read more in detail about what is the stance of repentance in Islam, you can do so here.