In general, Judaism does not approve of the purposeful damaging of the body or the making of permanent marks. Thus, tattoos, branding, and the punching of holes through the body are not generally accepted.
Earrings are a special case.
The Torah does record that wearing an earring in the upper cartilage of the ear is a mark of slavery, and thus is not allowed. Wearing an earring in the fleshy lobe of the ear has a long history and has been allowed, but only for women. There is also a history of Haredi Jews wearing earrings, and there are records of Chassidic Jews wearing earrings to prevent or cure poor eyesight.