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