How does divorce work in Islam?

ٱلطَّلَٰقُ مَرَّتَانِ ۖ فَإِمْسَاكٌۢ بِمَعْرُوفٍ أَوْ تَسْرِيحٌۢ بِإِحْسَٰنٍۢ ۗ وَلَا يَحِلُّ لَكُمْ أَن تَأْخُذُوا۟ مِمَّآ ءَاتَيْتُمُوهُنَّ شَيْـًٔا إِلَّآ أَن يَخَافَآ أَلَّا يُقِيمَا حُدُودَ ٱللَّهِ ۖ فَإِنْ خِفْتُمْ أَلَّا يُقِيمَا حُدُودَ ٱللَّهِ فَلَا جُنَاحَ عَلَيْهِمَا فِيمَا ٱفْتَدَتْ بِهِۦ ۗ تِلْكَ حُدُودُ ٱللَّهِ فَلَا تَعْتَدُوهَا ۚ وَمَن يَتَعَدَّ حُدُودَ ٱللَّهِ فَأُو۟لَٰٓئِكَ هُمُ ٱلظَّٰلِمُونَ ﴿٢٢٩﴾

Divorce must be pronounced twice and then (a woman) must be retained in honour or released in kindness. And it is not lawful for you that ye take from women aught of that which ye have given them; except (in the case) when both fear that they may not be able to keep within the limits (imposed by) Allah. And if ye fear that they may not be able to keep the limits of Allah, in that case it is no sin for either of them if the woman ransom herself. These are the limits (imposed by) Allah. Transgress them not. For whoso transgresseth Allah's limits: such are wrong-doers.

يَٰٓأَيُّهَا ٱلنَّبِىُّ إِذَا طَلَّقْتُمُ ٱلنِّسَآءَ فَطَلِّقُوهُنَّ لِعِدَّتِهِنَّ وَأَحْصُوا۟ ٱلْعِدَّةَ ۖ وَٱتَّقُوا۟ ٱللَّهَ رَبَّكُمْ ۖ لَا تُخْرِجُوهُنَّ مِنۢ بُيُوتِهِنَّ وَلَا يَخْرُجْنَ إِلَّآ أَن يَأْتِينَ بِفَٰحِشَةٍۢ مُّبَيِّنَةٍۢ ۚ وَتِلْكَ حُدُودُ ٱللَّهِ ۚ وَمَن يَتَعَدَّ حُدُودَ ٱللَّهِ فَقَدْ ظَلَمَ نَفْسَهُۥ ۚ لَا تَدْرِى لَعَلَّ ٱللَّهَ يُحْدِثُ بَعْدَ ذَٰلِكَ أَمْرًۭا ﴿١﴾ فَإِذَا بَلَغْنَ أَجَلَهُنَّ فَأَمْسِكُوهُنَّ بِمَعْرُوفٍ أَوْ فَارِقُوهُنَّ بِمَعْرُوفٍۢ وَأَشْهِدُوا۟ ذَوَىْ عَدْلٍۢ مِّنكُمْ وَأَقِيمُوا۟ ٱلشَّهَٰدَةَ لِلَّهِ ۚ ذَٰلِكُمْ يُوعَظُ بِهِۦ مَن كَانَ يُؤْمِنُ بِٱللَّهِ وَٱلْيَوْمِ ٱلْءَاخِرِ ۚ وَمَن يَتَّقِ ٱللَّهَ يَجْعَل لَّهُۥ مَخْرَجًۭا ﴿٢﴾

O Prophet! When ye (men) put away women, put them away for their (legal) period and reckon the period, and keep your duty to Allah, your Lord. Expel them not from their houses nor let them go forth unless they commit open immorality. Such are the limits (imposed by) Allah; and whoso transgresseth Allah's limits, he verily wrongeth his soul. Thou knowest not: it may be that Allah will afterward bring some new thing to pass. Then, when they have reached their term, take them back in kindness or part from them in kindness, and call to witness two just men among you, and keep your testimony upright for Allah. Whoso believeth in Allah and the Last Day is exhorted to act thus. And whosoever keepeth his duty to Allah, Allah will appoint a way out for him,

وَٱلْمُطَلَّقَٰتُ يَتَرَبَّصْنَ بِأَنفُسِهِنَّ ثَلَٰثَةَ قُرُوٓءٍۢ ۚ وَلَا يَحِلُّ لَهُنَّ أَن يَكْتُمْنَ مَا خَلَقَ ٱللَّهُ فِىٓ أَرْحَامِهِنَّ إِن كُنَّ يُؤْمِنَّ بِٱللَّهِ وَٱلْيَوْمِ ٱلْءَاخِرِ ۚ وَبُعُولَتُهُنَّ أَحَقُّ بِرَدِّهِنَّ فِى ذَٰلِكَ إِنْ أَرَادُوٓا۟ إِصْلَٰحًۭا ۚ وَلَهُنَّ مِثْلُ ٱلَّذِى عَلَيْهِنَّ بِٱلْمَعْرُوفِ ۚ وَلِلرِّجَالِ عَلَيْهِنَّ دَرَجَةٌۭ ۗ وَٱللَّهُ عَزِيزٌ حَكِيمٌ ﴿٢٢٨﴾

Women who are divorced shall wait, keeping themselves apart, three (monthly) courses. And it is not lawful for them that they should conceal that which Allah hath created in their wombs if they are believers in Allah and the Last Day. And their husbands would do better to take them back in that case if they desire a reconciliation. And they (women) have rights similar to those (of men) over them in kindness, and men are a degree above them. Allah is Mighty, Wise.