There are different kinds of punishments in Islam, there are hodouds, which are fixed by God, they apply on theft, zina, Hirabah , drinking alcohol, accusing someone of illicit sex but failing to present enough evidence (not sure if there are more sins in this category).
There are also Qisas and Diyya, which apply on murder, or when the criminal deals body injuries to the victim.
From the Quran:
وَكَتَبْنَا عَلَيْهِمْ فِيهَا أَنَّ النَّفْسَ بِالنَّفْسِ وَالْعَيْنَ بِالْعَيْنِ وَالْأَنْفَ بِالْأَنْفِ وَالْأُذُنَ بِالْأُذُنِ وَالسِّنَّ بِالسِّنِّ وَالْجُرُوحَ قِصَاصٌ ۚ فَمَنْ تَصَدَّقَ بِهِ فَهُوَ كَفَّارَةٌ لَهُ ۚ وَمَنْ لَمْ يَحْكُمْ بِمَا أَنْزَلَ اللَّهُ فَأُولَٰئِكَ هُمُ الظَّالِمُونَ
And We ordained for them therein a life for a life, an eye for an eye, a nose for a nose, an ear for an ear, a tooth for a tooth, and for wounds is legal retribution. But whoever gives [up his right as] charity, it is an expiation for him. And whoever does not judge by what Allah has revealed – then it is those who are the wrongdoers. (Verse 5:45)
For example, in case of murder, the family of the victim can choose between forgiving the victim, or killing him (of course, it's the governor's job), in case of body injuries, the victim can do the same body injuries to the sinner.
If the sin does not fall in the previous two categories, then there is Tazir تعزير (the word Tazir means to educate in form of a reprimand or reproach), which is a punishment that is not stated in the Quran or the Sunna, it's man-made, the governors are free to choose the appropriate one (depending on the sin), it can be prison, a fine, flogging, banishment, and execution in some cases, more information about Tazir, with examples of sins that Tazir punishes can be found here.
Only Tazir is man-made.