“Elf magic isn’t like wizard’s magic, is it? ...I mean, they can Apparate and Disapparate in and out of Hogwarts when we can’t.”
-- Ron Weasley (DH10)
House-elves have a very powerful magic all their own, magic which requires none of the types of focusing tools (wands or words) that Wizard magic requires. Dobby used his magic to enchant a Bludger, to close the magic portal that leads to platform nine-and-three-quarters, to interfere with Owl Post and steal someone’s letters, and to knock Lucius Malfoy down a flight of stairs. He also can disappear at will, which would seem to be a form of Apparition, but a form which can be done within Hogwarts, where, as Hermione constantly reminds Harry and Ron, normal Wizarding Apparition is simply not possible. All of these are very powerful charms.
Unfortunately, as long as they’re enslaved, they aren’t likely to be using this magic for anything but their mundane work. Hermione Granger is working to bring enlightened thinking to the house-elves at Hogwarts, so far with very little success. It would be interesting indeed for the Wizarding world if she would succeed in convincing these creatures to stand up for themselves.