Like most modern religions, moderate Islam tends to permit (tolerate might be a better word) tattoos and body piercings, while stricter interpretations of the faith tend to ban it. It should be noted that there are of course many Muslim regions where tattooing is culturally accepted.
Shari'ah (Islamic law) warns against Muslah (disfiguring) of the body, as well as "unnecessary interference, alteration, and mutilation of Allah's creation." Now, this doesn't strictly speaking ban tattoos, but under most interpretations it does. Sheik Yusuf Al-Qaradawi writes,
- Islam denounces excesses in beautifying oneself when it alters the physical features that Allah created him with. The Glorious Qur'an considers such alterations as inspired by Satan, who "...will command them (his devotees) to change what Allah has created..." (An-Nisa': 119)
The prophet is said however to have specifically banned many cultural practices of the time. In Sahih Bukhari Abu Huraira writes that Muhammad forbade tattooing, and Abdullah continues,
- Allah has cursed those women who practice tattooing and those who get themselves tattooed, and those who remove their face hairs, and those who create a space between their teeth artificially to look beautiful, and such women as change the features created by Allah.
That said, in many Arabic areas that are devoutly Muslim, tattooing is quite common. In addition, tattooed people are welcome to convert to Islam (their tattoo sins are forgiven in the process).
Body piercing is less clearly banned, although many Muslims believe that the bans on unnecessary interference and mutilation of the body clearly bans most elective mods in the process. After all, the Koran (An-Nisa': 119) specifically warns that Satan will try and trick humans into body modification:
- "And I will surely lead them astray, and arouse desires in them, and command them and they will cut the cattle's ears, and I will surely command them and they will change Allah's creation." Whoever chooses the Devil for a friend instead of Allah is assuredly a loser, and his loss is manifest.
Women are permitted to get ear piercings, and under certain circumstances (when the cultural norms of the area permit it without behaving strangely), other piercings may be permitted. However, just because a woman can do it doesn't mean a man can do it. The Prophet was quite clear that different rules apply across genders ("Allah has cursed men who copy women as well as women who copy men"), and similar edicts applied to borrowing customs (including piercing) from other cultures. So in terms of a short answer, body piercing isn't allowed either, but it's definitely a more grey area if you live in a region where it's "normal."
Cultural circumcisions are also permitted, although FGM (often linked as a Muslim practice) is generally considered banned under Shari'ah as well and is more a remnant of the cultural practices of the region than the faith.