One of those exquisite mysteries of scripture. They are only vaguely mentioned in two or three places.
Probably the clearest mention (and it's vague) is found in Exodus 28 verse 30. They are to be worn in the breastplate, over the heart by Aaron the High Priest when he stand before the Lord. (i.e. before the Ark of the Covenant in the Holy of Holies). They appear to be some sort of oracular devices.
I believe they are also mentioned (vaguely) in the Book of Nehemiah. After the walls and Temple are rebuilt, the elders commission a search to find the Priest to wear the Urim and Thummin. I believe the Priest they find is Ezra.
I know the Urim is mentioned somewhere else, I just can't remember where.
I hope this helps.