Dumbledore wanted to make Tom Riddle a better person, and he knew that Tom could do magic. He could tell that Tom Riddle had evil thoughts, but Dumbledore gave Tom Riddle another chance.
It might have been the property of one of the four founders of Hogwarts, but no one really knows. The mirror of Erised shows you what you want, and the one who looks into this mirror will only see himself exactly as he is.
