Short answer
A permanent tattoo is not. Islam teaches that the body is a trust from God (﷾), not something we own and may mark as we like. The Prophet (ﷺ) forbade the permanent tattoo, and Muslim scholars agree on it, for men and women alike. The Qur'an counts defacing what God (﷾) made among the things Satan calls people to. Islam is not against beauty: a design that fades, such as henna, is allowed. And the rule is older than Islam. The Torah says: "nor print any marks upon you". If you already have a tattoo, it is no barrier to Islam. Islam wipes out what came before it, and no one is asked to harm themselves to remove one.
Full explanation
What the Prophet (ﷺ) said
Two Companions report it. Ibn Umar says the Prophet (ﷺ) said that Allah (﷾) has cursed the woman who tattoos and the woman who has it done. Ibn Mas'ud says Allah (﷾) has cursed those who tattoo and those who have it done. He names them among others who change the creation of Allah (﷾) for beauty. A woman asked him where this was in the Qur'an. He pointed to the verse that tells believers to take what the Messenger (ﷺ) gives them, and refrain from what he forbids. A curse falls only on a wrong act. And the jurists apply the rule to men as well as women.
Ibn Umar reports that the Messenger of Allah (ﷺ) said that Allah has cursed the woman who adds false hair or has it added, and the woman who tattoos or has herself tattooed.
Abdullah ibn Mas'ud said that Allah has cursed the women who tattoo and those who have themselves tattooed, those who pluck the hair of the face and those who file gaps between their teeth for beauty, changing the creation of Allah. He asked why he should not curse those whom the Prophet (ﷺ) cursed, and pointed to the verse: what the Messenger gives you, take; and what he forbids you, refrain from (59:7).
“What Allah has bestowed on His Messenger (and taken away) from the people of the townships,- belongs to Allah,- to His Messenger and to kindred and orphans, the needy and the wayfarer; In order that it may not (merely) make a circuit between the wealthy among you. So take what the Messenger assigns to you, and deny yourselves that which he withholds from you. And fear Allah; for Allah is strict in Punishment.”
A permanent tattoo, made by piercing the skin with a needle so that blood comes out and filling the wound with dye, is forbidden: the jurists agree on it, on the evidence of Ibn Mas'ud's report in al-Bukhari and Muslim, since a curse falls only on a blameworthy act. It quotes the Maliki al-Adawi (the prohibition is for men and women alike, and is a major sin) and the Shafi'i al-Khatib al-Shirbini (tattooing is forbidden). A temporary design made with dyes that fade after a short time counts as permitted adornment, not as the forbidden tattoo; it quotes al-Qurtubi that the prohibition concerns what lasts, while what does not last, such as kohl, scholars allow to women.
Changing what God (﷾) made
The Qur'an tells of Satan's promise to lead people astray. Among the things he says he will order them to do is "to deface the (fair) nature created by Allah". Ibn Mas'ud names the tattoo among the changes the Prophet (ﷺ) forbade. A permanent mark cut into the skin with ink is such a change. Islam asks the body to be kept as God (﷾) made it, as a trust. It is lent to us for a lifetime, and we will be asked how we used it.
“"I will mislead them, and I will create in them false desires; I will order them to slit the ears of cattle, and to deface the (fair) nature created by Allah." Whoever, forsaking Allah, takes satan for a friend, hath of a surety suffered a loss that is manifest.”
Abdullah ibn Mas'ud said that Allah has cursed the women who tattoo and those who have themselves tattooed, those who pluck the hair of the face and those who file gaps between their teeth for beauty, changing the creation of Allah. He asked why he should not curse those whom the Prophet (ﷺ) cursed, and pointed to the verse: what the Messenger gives you, take; and what he forbids you, refrain from (59:7).
Not against beauty
What Islam forbids is the mark that lasts: a needle that cuts the skin, and ink sealed into the wound for life. Scholars of the different schools agree on this. A design that fades after a short time is different. Henna and other dyes that wash away count as adornment. As the Qur'an commentator al-Qurtubi put it, the prohibition concerns what lasts. So Islam does not ask people to give up beauty, only to keep the body as God (﷾) made it.
A permanent tattoo, made by piercing the skin with a needle so that blood comes out and filling the wound with dye, is forbidden: the jurists agree on it, on the evidence of Ibn Mas'ud's report in al-Bukhari and Muslim, since a curse falls only on a blameworthy act. It quotes the Maliki al-Adawi (the prohibition is for men and women alike, and is a major sin) and the Shafi'i al-Khatib al-Shirbini (tattooing is forbidden). A temporary design made with dyes that fade after a short time counts as permitted adornment, not as the forbidden tattoo; it quotes al-Qurtubi that the prohibition concerns what lasts, while what does not last, such as kohl, scholars allow to women.
If you already have one
A tattoo is not a barrier to Islam. When Amr ibn al-As asked to be forgiven before he became Muslim, the Prophet (ﷺ) told him that Islam wipes out everything that came before it. Scholars say an old tattoo may be removed if that is safe, and may be left if removing it would cause harm. Most say prayer with it is valid, and no sin remains after repentance. So no one needs to wait, or to hide, because of a tattoo. No Muslim should shame anyone for one. The page "How do I handle the 'haram police' as a new Muslim?" says more.
Amr ibn al-As, near death, said that before Islam no one was more hateful to him than the Prophet (ﷺ), and he had wished to kill him. He told how he came to the Prophet to accept Islam. He asked the Prophet to stretch out his right hand so he could pledge, then drew his own hand back. Asked why, he said he wanted a condition: that he be forgiven. The Prophet asked whether he did not know that Islam wipes out everything that came before it.
A permanent tattoo that traps blood under the skin is forbidden by agreement of the jurists, and the one who had it done must repent. It must be removed if that brings no harm. If specialists judge that removal would cause harm, it may be left, prayer with it is valid in the view of most jurists, and no sin remains on its owner after repentance.
The Torah said the same
The rule is older than Islam. The Law of Moses says: "Ye shall not make any cuttings in your flesh for the dead, nor print any marks upon you: I am the Lord." Jesus said he had not come to destroy the law. Through the Prophet (ﷺ), Islam keeps this rule for every believer.
“Ye shall not make any cuttings in your flesh for the dead, nor print any marks upon you: I am the Lord.”
“Think not that I am come to destroy the law, or the prophets: I am not come to destroy, but to fulfil.”
Perspectives
A tattoo harms no one
Many people see a tattoo as harmless self-expression: art, a memory, or a mark of who they are. It is their own body, they say, so forbidding it makes no sense.
A tattoo harms no one, so cursing it is out of all proportion.
Rebuttal
Islam does not say the body is ours to change as we like. It is a trust from God (﷾), and the Qur'an counts defacing what He made among the things Satan calls people to. A believer gives it up because God (﷾) and His Messenger (ﷺ) decided it (33:36). And the door stays open: Islam wipes out what came before it.
Christians are not under that law
Many Christians accept that Leviticus forbids marks on the body. But they hold that the Law of Moses was for Israel, and that Christians are not under it.
Paul says Christians are not under the law but under grace, so Leviticus 19:28 does not bind them.
Rebuttal
Jesus said: "Think not that I am come to destroy the law, or the prophets." The words about grace are Paul's, not his. Through the Prophet (ﷺ), Islam keeps the old rule against marking the body for every believer.
The Islamic perspective
The body is a trust from God (﷾). A permanent tattoo changes what He made, so Islam forbids it, and its scholars agree. A design that fades, such as henna, is allowed. A tattoo made before Islam is no barrier: Islam wipes out what came before, and one that cannot safely be removed may be left.
The Prophet (ﷺ) cursed the one who tattoos and the one who has it done.
The jurists agree that a permanent tattoo is forbidden, for men and women alike.
An old tattoo should be removed if that is safe. If removing it would cause harm, it may be left, and most jurists say prayer with it is valid.
Islam wipes out what came before it.