It’s not Jewish to get a tattoo. This is not withstanding the range of “Jewish” images that are available like stars of David, kabbalistic sephirot or images of dancing Chasidim.
The Torah is clear about this, stating: “You shall not make gashes in your flesh for the dead, or incise any marks on yourselves: I am the Lord” (Leviticus 19:28). The Mishnah teaches that it is the lasting and permanent nature of tattooing which makes it a culpable act (Makkot 3:6). This teaches an important lesson about tattoos: although at a certain age it can seem like a good idea to imprint ourselves indelibly with the image of our favourite pop idol, or even of a star of David or dancing Chasid, in 50 years’ it may not seems so cool, but by then we’ll be stuck with it (and laser surgery is painful and expensive).
Rabbi Simeon ben Judah says that it is the inclusion of a god’s name which makes it a culpable act; the association of tattooing with idolatry is the overriding factor. But we follow the opinion that any permanent deliberate disfiguration to our God-given bodies is forbidden.