'Azl, or coitus interruptus (withdrawing before ejaculation) and barrier methods:
"With regard to ‘azl (coitus interruptus), or withdrawing during intercourse, the correct scholarly view is that there is nothing wrong with it, because of the hadeeth of Jaabir (may Allah be pleased with him): “We used to practise ‘azl at the time when the Qur’an was being revealed” – i.e., at the time of the Prophet (peace and blessings of Allah be upon him). If that action had been haram [forbidden], the Prophet (peace and blessings of Allah be upon him) would have forbidden it. But the scholars say that one should not engage in ‘azl with a woman except with her permission, because she has the right to have children. Moreover, withdrawing without her permission diminishes her pleasure, so not asking her permission causes her to lose out on pleasure and on the possibility of having children. Hence we state the condition that this may only be done with her permission."
From Fataawa al-Shaykh Muhammad ibn ‘Uthaymeen.
Condoms and other barrier methods are viewed in the same light a the withdrawal method, but again, both partners must agree to using them.
Birth Control Pills:
A woman should not use birth control pills, unless the following conditions are met:
1- She should need to use them, for example if she is ill and cannot cope with a pregnancy every year, or she is physically unfit, or there is some other reason that getting pregnant every year may harm her.
2- Her husband should give his permission, because the husband has the right to have children. There must also be consultation with the doctor, to find out whether these pills are harmful or not.
If these two conditions are met, there is nothing wrong with taking these pills, but that should not be on a permanent basis, because that means preventing having children.
IUD, Mirena, and other coil methods:
Using the coil is permissible subject to two conditions:
The first condition is that it should not cause harm to the woman.
The second condition is that her husband should give his permission for that.
Permanent Birth Control (Vasectomy, Tubals, etc):
Shaykh ‘Abd al-‘Azeez ibn Baaz [A prominent scholar] (may Allah have mercy on him) was asked:
What is the ruling on removing the uterus in order to avoid having children for medical reasons which are either present, or may occur in the future and have been predicted by medical and scientific means?He answered: if that is necessary, then it is OK, otherwise it should not be done, because the Lawgiver urges us to have children and promotes that in order to increase the size of the ummah [nation]. But if there is a necessary reason then it is OK, just as it is permissible to use means of contraception for a limited time for a legitimate shar’i reason. (9/434)
In conclusion, birth control is a personal decision for each couple, and as long as the method causes no harm to the individual, and they do have a legitimate reason for using it, they can. Permanent birth control is frowned upon, but is allowed if the mother's health is in danger during pregnancy.
Source: Islam Q&A