The word קעקע as used in Vayikra 19:28 probably originally meant brands, of the type used to mark criminals or, in some neighboring cultures, as mementi mori (marks made in the flesh to remember the dead), and indeed, the verse begins by saying that what is being referred to is שרט לנפש, which is to say, "marks for the dead." Our Rabbis, in the Talmud, use the verse as a prooftext for prohibiting certain kinds of tattoos, also.
What the halachah originally prohibited was marks in the flesh dedicated to the worship of foreign gods, or which replicated one of our own divine names of God which we customarily do not write or say except in prayer or sacred books.
Over time, however, the custom grew universal in Jewish communities to not make any voluntary marks in our flesh at all, whether tattoos or brands or scarification, or anything else. Some rabbis cited the Rabbinic commandments not to harm oneself, as well; or cited prohibitions about practicing the ways of idol worshippers; or other various customs or halachot as support for the strengthening of this custom.
In Orthodox Judaism, the halachic maxim minhag yisrael k'halachah hee is commonly understood to mean that any custom practiced in a Jewish community for a long enough period takes on the force of halachah for that community; and any custom practiced in a plurality of Jewish communities for a long enough period takes on the force of halachah for everyone.
Therefore, Orthodox Judaism completely forbids getting tattoos voluntarily. Obviously, the prohibition does not apply to tattoos not gotten voluntarily, like those forced upon inmates of the concentration camps in the Shoah. Nor does it require the removal by converts of tattoos acquired before conversion. But it is an absolute blanket prohibition for all Jews, according to Orthodox understandings of halachah.