What function did these tattoos serve? Who got them and why?
Because this seemed to be an exclusively female practice in ancient Egypt, mummies found with tattoos were usually dismissed by the (male) excavators who seemed to assume the women were of "dubious status," described in some cases as "dancing girls." The female mummies had nevertheless been buried at Deir el-Bahari (opposite modern Luxor) in an area associated with royal and elite burials, and we know that at least one of the women described as "probably a royal concubine" was actually a high-status priestess named Amunet, as revealed by her funerary inscriptions.
And although it has long been assumed that such tattoos were the mark of prostitutes or were meant to protect the women against sexually transmitted diseases, I personally believe that the tattooing of ancient Egyptian women had a therapeutic role and functioned as a permanent form of amulet during the very difficult time of pregnancy and birth. This is supported by the pattern of distribution, largely around the abdomen, on top of the thighs and the breasts, and would also explain the specific types of designs, in particular the net-like distribution of dots applied over the abdomen. During pregnancy, this specific pattern would expand in a protective fashion in the same way bead nets were placed over wrapped mummies to protect them and "keep everything in." The placing of small figures of the household deity Bes at the tops of their thighs would again suggest the use of tattoos as a means of safeguarding the actual birth, since Bes was the protector of women in labor, and his position at the tops of the thighs a suitable location. This would ultimately explain tattoos as a purely female custom.
Who made the tattoos?
Although we have no explicit written evidence in the case of ancient Egypt, it may well be that the older women of a community would create the tattoos for the younger women, as happened in 19th-century Egypt and happens in some parts of the world today.
What instruments did they use?
It is possible that an implement best described as a sharp point set in a wooden handle, dated to c. 3000 B.C. and discovered by archaeologist W.M.F. Petrie at the site of Abydos may have been used to create tattoos. Petrie also found the aforementioned set of small bronze instruments c. 1450 B.C.—resembling wide, flattened needles—at the ancient town site of Gurob. If tied together in a bunch, they would provide repeated patterns of multiple dots.