This results from the phenomenon called "electromagnetic induction" sometimes called the "Faraday Effect." When any conductor like aluminium is exposed to a changing magnetic field, electric currents are induced. (This is how electricity is produced in generators.) These induced currents are called "eddy currents" and tubes like the aluminium one you describe are often called "eddy current tubes."

As the magnet falls in the aluminium tube, the tube experiences a changing magnetic field produced by the falling magnet. This induces electric currents around the tube that tend to slow down the magnet's fall. Such tubes are often used in physics classes to demonstrate electromagnetic induction. The same effect can occur with any permanent magnet, but the effect is strongest with the strongest magnets, i.e., with rare earth magnets.