It is Biblically forbidden to make a Tattoo on one’s skin.
Definition-How its done: The tattoo [that is forbidden by the Torah] is made by making a cut in one’s skin and then filled it with sand or ink or other pigments that leave a mark. Alternatively, one first marks an image on the skin using a pigment and then cuts open the skin for the ink to enter into its pours. [Today tattoos are made through first sketching an image onto the skin and then using a fine needle that is filled with ink to puncture the skin and enter the ink. Thus the cutting and filling is done simultaneously.]
Where on the skin? The Biblical prohibition against making a tattoo applies to anywhere on the skin.
What writing is forbidden? The Biblical prohibition against making a tattoo applies to any type of writing. [It however only applies to letters of a language and does not apply to making a mere mark or line and the like, although doing so is Rabbinically forbidden.]
Having someone else do it: If someone else makes a tattoo on one’s skin then if he assisted him in the tattooing he is liable. If he did not assist him in making the tattooing then he is exempt from Biblical liability. Nevertheless, it remains Rabbinically forbidden.
Why did the Torah prohibit tattoos?
The act of tattooing is rooted in the act of idol worship, in which the worshipers would tattoo their god onto their skin, thus showing their subordination to him.
May one write a tattoo onto the skin of a gentile?
May one with a tattoo be buried in a Jewish cemetery?
A Jew that has a tattoo must be buried in a Jewish cemetery just like any other Jew. Despite the common misconception, there is no Halachic source that bars a Jew with a tattoo, or a Jew who has committed any sin, from being buried in a Jewish cemetery, nor is this the common practice.
 Shulchan Aruch 180; Mishneh Makos 21a
 Chinuch Mitzvah 253
 Michaber 180/1; Mishneh Makos 21a that one must both cut and fill it with ink
 Shach 180/1; Bach
 Shach 180/2
 Shach 180/2
 Meil Tzedaka 31, brought in Pischeiy Teshuvah 180/1
 Michaber 180/2 and 11
 Shach 180/4
 Shach 180/6; Chinuch Mitzvah 253
 Noda Beyehuda Tinyana, brought in Pischeiy Teshuvah 180/2
 See Yoreh Deah 362/5; Sanhedrin 47a that one is not to bury a Rasha next to a Tzaddik, which is the source we find for the custom of having a Jewish cemetery that is free of gentiles. It would likewise based on this be prohibited to bury a Rasha who is a Jew near a gentile, and hence one with a Tattoo may not be buried with gentiles even though he has committed a sin. See Igros Moshe 147 that even a Jew who is married to a gentile must be buried in a Jewish cemetery.