Trains that levitate thanks to the use of a magnetic field are called “Maglevs”, which stands for “Magnetic Levitation trains”. Maglevs can potentially be faster and quieter than classical wheel trains, such as the TGV. Today (in 2011), there are mainly two competing technologies to build a Maglev: electrodynamic sustentation (EDS) and electromagnetic sustentation (EMS). EDS is a technique that uses superconductors. This technique relies on inducted currents; similar to the currents created when you make a magnet move above a metal plate.
When you move a magnet next to a metal, even a non-magnetic metal such as copper, currents appear in this metal. The presence of these currents has been known for a long time (Lenz law) and can be explained by the same laws used for magnetic and electric fields: they are called “Eddy currents” (“Foucault currents in French), or “inducted currents”.
There is a simple experiment to prove the existence of such currents: a strong magnet is dropped in a copper tube (https://www.youtube.com/watch?v=30oPZO_z7-4). The magnet then falls very slowly: reacting to the fall of the magnet, Eddy currents appear on the tube and create a magnetic field that applies a force on the magnet and slows it down.
A moving magnetic field can hence produce inducted currents that, in reaction, will produce a second magnetic field interacting with the first one. It is this force that lifts the Maglev.
Un champ magnétique qui se déplace peut donc créer des courants induits, qui vont en réaction créer un second champ magnétique qui va interagir avec le premier. C’est cette force qui va permettre de soulever le Maglev.