Some view tattoos as a rebellion and an attack on conventional American dress. In this way of thinking, tattoos are a way to give the middle finger to society or to people (maybe parents?) that you disagree with.
This is related to viewing tattoos as self-expression. While some express themselves (and their aforementioned middle finger) with tattoos in the form of rebellion, others use the permanent markings to share bits of themselves with the world. I've talked to friends who can explain the meaning of each minute detail of their tattoos and what it means to them and their life.
Some, however, get tattoos simply because it appeals to them, perhaps getting a common tattoo shape or style (like a feather, an infinity sign, a tribal tattoo or an Ed Hardy-esque dagger) because it looks appealing.
The way different people think about tattoos is interesting and occasionally contradictory. I mentioned in class how my mom, in most ways a feminist, abhors tattoos on women (though her son and husband both have tattoos). However some people find tattoos have the exact opposite effect. I, for example, find tattoos extraordinarily attractive, particularly when they are well done, are colorful and have some meaning to the wearer.
Personally, I view tattoos as a work of art. While there are many examples of bad tattoos, but good ones can be jaw-dropping. Here are a few of my favorites:
These tattoos show very well what skill goes into making a high-quality tattoo. These pieces would be difficult to create simply using a pen and pencil, and these tattooists managed to put them on tattoo them on humans, an awesome feat. These tattoos hint at a world of art that hasn't yet been tapped by the formal art world. Incorporating tattoos into common art would open up an entirely new medium to the art world.
Let's just hope it doesn't develop to the point of the MOMA in Futurama: