The church teaches and discourages having tattoos and body piercings. Pres Hinckley gave a memorable talk to the youth and young adults of the church back in Nov. 2000. It was reprinted in the Ensign Magazine in Jan. 2001. 
 
Here is an excerpt:  
 
"I promise you that the time will come, if you have tattoos, that you will regret your actions. They cannot be washed off. They are permanent. Only by an expensive and painful process can they be removed. If you are tattooed, then probably for the remainder of your life you will carry it with you. I believe the time will come when it will be an embarrassment to you. Avoid it. We, as your Brethren who love you, plead with you not to become so disrespectful of the body which the Lord has given you." 
 
"May I mention earrings and rings placed in other parts of the body. These are not manly. They are not attractive. You young men look better without them, and I believe you will feel better without them. As for the young women, you do not need to drape rings up and down your ears. One modest pair of earrings is sufficient." 
 
"I mention these things because again they concern your bodies." 
 
"How truly beautiful is a well-groomed young woman who is clean in body and mind. She is a daughter of God in whom her Eternal Father can take pride. How handsome is a young man who is well groomed. He is a son of God, deemed worthy of holding the holy priesthood of God. He does not need tattoos or earrings on or in his body. The First Presidency and the Quorum of the Twelve are all united in counseling against these things." 
 
Here is a link to the talk: 
Ensign Talk

Our bodies are temples. Temples that God created himself.  
Just like with a temple building, you would never mark it, graffiti it, or drill holes into it. The same is with your body. You should never mark your body or put holes into your body. 
I have a friend who is an evangelist and she decided to get a tattoo on the small of her back with a cross and a Psalms scripture on it. She tried to explain to me that it was acceptable to have a tattoo if it expressed your belief and devotion to God. I then explained to her that “Man looketh on the appearance, but God looketh upon the heart.” He knows if you are devoted to Him or not, through your faith, your words, and actions. I am not sure if she took to heart what I said to her but, I still stand by what I said. 
Not to mention, think about what your body would look like when you’re 50+? 
A lot (not all) of tattoos or piercings are also caused by a phase in life. So what if you finally decided to get it removed? All the money and pain to get it removed! 
Our bodies are more of a privilege sent from God, so in reality, our bodies are not essentially "ours." What we do with our bodies is under the order of Him, not only with tattoos or piercings, but with what we take into our bodies (Word of Wisdom), what comes out of our body (clean language), and relations that are saved only for marriage in order to multiply and replenish the earth. 
 
I hope this has helped and brought some consideration. 
Thanks! 
:)

my mormon friend said the body is eternally god's and she's just using it at the moment. I replied "except the ears. the ears are yours," necause she has ear piercings. explain.