Tattoos are discussed here.
Piercings can be an issue for two primary reasons: 1) keeping your body intact and in its natural form and 2) because piercings used to indicate that a person (particularly a male) was a slave.
In general today, piercings don't appear to be an issue for conversion candidates except for social reasons. Ear piercings, even multiple ones, probably won't be brought up. If you're a "young" woman (under 30 or so), nose piercings generally won't be an issue. A belly button ring likely won't be an issue because if you're dressing tzniusly, no one should know it's there. I even had/have an unusual ear piercings (tragus), and in my old community, no one thought anything of it. Of course, know your community, and there are communities with standards considerably stricter.
However, other facial piercings or tongue piercings will likely bring a reaction from a beit din. They will warn you that orthodox society doesn't look favorably on them and it could hamper your integration into the community.
There is also a practical consideration: the mikvah. Can you remove any piercing in order to go to the mikvah? In my case, I had a skin piercing that had to be removed once I knew I would be heading to the mikvah for my conservative conversion. The nape of my neck was pierced (very pretty and feminine, if you ask me!). Skin piercings require going back to a piercer to reinsert them, so I felt that it was easier for everyone if I let the piercing grow up. I miss it a little, but it's been worth it. The fact that (Gd willing) one day I'll be a regular mikvah user was certainly a consideration to me.
But remember, when in doubt, ask. However, be prepared that you may get an answer you don't like and/or don't personally agree with.