The only materials that exhibit strong and easily observable forces with ordinary magnetic fields are ferromagnetic materials (like iron) and ferrimagnetic materials (like many iron oxides). However, no materials are truly "nonmagnetic". Most are either paramagnetic (weakly attracted to a magnetic field) or diamagnetic (weakly repelled by a magnetic field), but with forces so weak that they are not easily observed. However, if you have an extremely strong magnetic field and a strong gradient in that field, as can be obtained at the edge of a high-field electromagnet, you can produce a force strong enough to balance gravitational forces and produce diamagnetic levitation. Diamagnetic materials that have been levitated in this way include drops of water and even living things like insects and small frogs.
[Editor adds: You can learn more about this at the University of Nijmegen, The Netherlands, at their Frog that Learned to Fly webpage. Be sure to check the videos.
The explanation given is that everything is made of atoms with spinning electrons. Each atom is like a little magnet, but in "non-magnetic" materials these are not all lined up. When a very strong magnetic field is applied, they all line up with their poles opposite to the applied magnetic field, making them diamagnetic. You need a magnet about 100 or 1000 times more powerful than a household magnet.]