Om and Aum are the same thing; it's just written two different ways. Think of "Om" as the phonetic spelling and "Aum" as the actual spelling. The sound of "Aum" - the slow, calming chant many associate with the word, is the sound that was made when all of creation came into existence. The essence of the universe and all creation, wrapped up in one unimaginable and indescribable aggregate (for lack of a better term), is known as Brahman. The "Om" represents the four divine states of Brahman - metta (loving kindness), karuna (compassion), mudita (sympathetic joy), and upekkha (equanimity). Brahman is actually a core belief system in both Hinduism and Buddhism, both of which use Om regularly in their daily life. (Feel free to research further on this with the links below - it's quite fascinating.)
So, just the sound of "Aum" is so beyond human comprehension that it boggles the mind, and then there is the symbol as well. Just as Brahma is the culmination of everything, so is the Om symbol. It's written in Sanskrit, and each part of the symbol has a very significant purpose. The two curls on top of each other (that sort of look like the number 3), with the downward curl that spirals out from the "3" shape, each represent a state of consciousness. The large, bottom curl stands for the normal state of being awake, which you probably are in right now. The curl above it stands for deep sleep, while the curl emanating out from the center of the two represents the dream state. Those are the states of consciousness that every human on earth experiences. But the dot and the open curve above that is what elevates the symbol to a much higher and more sacred meaning. The dot represents absolute consciousness; that's not the same as merely being awake, but it means being fully aware of yourself and everything around you. The open curve that cradles the dot represents an infinitely open mind, which is required to achieve that level of absolute consciousness.* (See note below)
As you can see, it's not just a soothing chant or a pretty symbol. If you plan to place an "Om" on your body, you need to understand and embrace all that it stands for.
Note: After my own personal research on this symbol, it occurs to me that the symbol in the tattoo seen here may actually be incorrect as it's missing the dream state curl to the right of the "3" shape. I'm not knowledgeable enough on the subject to know if this is an acceptable variation to those in the various faiths that hold the "Om" in great esteem. I'm going to continue doing some research and will hopefully have more information to add in coming weeks. There are, however, known acceptable variations: the Jain Om, the Tibetan Om, the Vedic Om, and the Sikh Ek Onkar.