Usually these little ivory sculptures -- called "Memento Moris" -- have a living person's face on one side and a grinning skull on the other. Most of the ones I've seen have some arrangement for stringing as pendants -- either an attached ring at the top or a vertical hole bored through them.
(Here's a closeup of the skull side of the example above.)
I don't have a lot of data on how these were actually used or worn -- they are almost always described as components of a rosary or paternoster, but I haven't seen any pictorial or documentary evidence of them being used that way. (Rosaries like the one in my first post are usually reconstructions.) I would be unsurprised to see them on, for instance, a watch chain or somewhere else that a small hanging decorative object would be used. These remind me strongly of the Japanese netsuke, similar little sculptures that have become a "hot" collectible art form.
Like any other expensive little accessory, these were probably worn as much to show off one's wealth and good taste as for any other reason. However they do have a serious spiritual purpose: as their name indicates, these are reminders that death comes to everyone ("memento mori" = "remember death") and that since it may be unexpected and sudden, being prepared for it spiritually is a good thing.
(Note that the left-hand one of the three below does not actually have a skull -- it has a man's face on one side and a woman's on the other. This makes me wonder whether some of these faces may be actual portraits of the owners rather than generic faces.)
I also suspect that, in the days before today's mass media, these gruesome little pieces catered to the same macabre taste as the popular "dance of death" murals -- the most popular one was in Paris -- and paintings and engravings showing a skeletal Death taking the hand of bishops, aristocrats, nuns, and ordinary people. Perhaps it's the same taste that leads modern people to watch vampire movies :)
(I've noticed worms and other creepy-crawlies on the "skull" side of several of these sculptures. This especially gruesome example has them on the "face" side as well. Eeeeeeeuuuuuuwwwwww!)
Posts in this series:
Death's head devotions
Skulls: the inside story
Skulls: the inside story, part 2
Skulls: the inside story, part 3
Voldemort, part 2
A skull of one's own
More living color