As the Minister for Wizard Public Relations Joanne Rowling states, Wizards and Witches need a wand to use magic. There are exceptions- after all, which one of us hasn't lost our temper with unfortunate magical consequences? But what about races unable to use wands? This is intended to give the Wizarding World a brief insight into a few other types of magic.
As a brief introduction, two apparently simple ideas must be defined. First, what is a wand? Every Witch or Wizard has one by their eleventh year, usually chosen specifically for its wielder. England's Ministry of Magic describes a wand as a 'mostly smooth length of wood from a Wand- quality wood tree, containing a core of an item from a magical creature'. [Legal terms for Legal Wizards, 107th edition, pg. 1953]. For this work, though, any item of one material with a different material core is considered to be a wand. (This includes the long-knives or trumpets of the Saint- Blessed Knights of Italy, or the labrys of the Artemisian Amazons.)
Second, what is magic? This has two popular explanations that start from the same place: magic is a force that powers charms, spells, potions, and lots of other things. The difference is that some believe that magic comes from and connects all living creatures while others believe that it comes from inside each Witch or Wizard. The true theory is left to the reader to decide.
Now, on to the creatures!
Most Wizards and Witches have heard of the Centaurs. These beasts- called such by their own request, not by any anti-Humanoid feelings- are able to use magic. This author has seen several herd members using what we would consider small magic. Instead of a wand, which the Ministry has outlawed for "beasts", Centaurs were using body movements. Each tail flick, hoof stamp, or hand movement appeared to be similar, but results were very different. The same hoof stamp and hand wave that lit the evening fires seemed to be the ones that lifted the colt from the gully.
How could these creatures use spells without a wand? The simplest explanation is that each Centaur is so simply magical that they do not need wands to focus their spells. Using the theory that magic comes from inside a Witch or Wizard, the best explanation is that the Centaur can use his or her own body to focus their magic. A Centaur's large size serves as either a collector of magic, or gives that much more area to direct magic through.
Next is a being as humble as Centaurs are aloof. The House Elf is a sight many Wizards have heard of if not seen. While the Ministry of Magic has also forbidden these beings the use of a wand, no one can deny that they use magic. Cooking, cleaning, and even defending their masters are among the many things House Elves have been seen to do.
Again, how could these beings use magic without a wand? One of the reasons is the same as for Centaurs: they are so completely magical that the focus of a wand is unnecessary. If the reader accepts that magical abilities come from within, a House Elf is the focus for their spells. For those that believe magic comes from without, perhaps a House Elf's overlarge eyes and ears can allow them so see and absorb this magic, in some sort of reverse radiator? Their over-long fingers can allow them to focus this magic, no matter which theory is believed. Even an ear twitch can be used to cast a charm as surely as a Witch can wave a wand.
House Elves can also be so attuned to their Home and Duties that they no longer need to have a focus for a spell. (After all, the older we get the easier it is to do spells without a wand.) Since very few have seen a House Elf doing magic outside of their Home, this might be true. Given the House Elf's long life- span, and the fact there are only so many ways to cook an egg or shine a boot, this would also explain their powerful, yet seemingly limited, charms.
One of the most confusing abilities of House Elves is that they are the only being, beast or spirit able to Apparate inside Hogwarts! How can this be when all others cannot? Perhaps this is because a House Elf's magic allows him or her to move through their Home faster than anyone can see. While technically not the same as Wizarding Apparition, it is close enough to call it that. They are able to harness their magic and step outside time as we know it to hurry to where they are needed, or to perform their chores with the speed they are famous for. Unfortunately, they cannot or will not explain what happens while they... Travel. Until one speaks up or brings a Wizard to along, it is the best explanation available.
Next are the Merfolk. These beings are very good at casting what we consider "small magic". Their charms deal with such things as lighting their shelters, scaring away or confusing trespassers, keeping their watery domain from freezing, and clearing away debris washed into their settlements. Wreckage from a collapsed bridge was cleared with no more than a wave of their finned hand. This author saw a gigantic octopus churn up a blinding cloud of ink and dirt, and the Merfolk Chieftain cleared away it away simply by pointing at it.
Their charms could be dismissed as using water currents, or intensifying them, much as a ripple grows larger as it flows away from the stone that caused it. This does not fit the facts though. If this were true, a charm to clear away debris would take the shelter with it! A 'hex' that led fishermen away by sweeping their boats away would drag the Merfolk along as well. Merfolk use true spells, just as much a well-crafted Charm as any Dry land Witch would use. The only difference between the Blasting Curse of a Hogsmeade Wizard and the similar Merfolk spell (it translates roughly to "SPLASHHurt you go away from me") is that a Merfolk will just as likely dive below its opponent's spell and cast its own from a different angle. Almost any spell that can be used in air can be used by a Merfolk, and any spell that can't has its own version. In three dimensions.
Finally, the most famous example of a magical creature: the Dragon. There are rumours that these creatures are able to use magic as well simply being magic. Unfortunately, this author was unable to confirm these rumours, since it is very difficult to take notes when one is running for your life.
However, this author would like to bring up one question that can only be answered by the fact that dragons can use magic. This is the ability of a dragon to hunt and capture prey. Because of their size, dragons must have some sort of ability to allow them to get close to their prey, since they tend to be quite noticeable.
Could this magic be something as simple as invisibility? Or perhaps they can fade like a chameleon, fitting into the background? It is possible, although even calling it 'magic' doesn't explain how a dragon could be overlooked. The closest explanation is a Confundus Charm. This would explain how dragons have the ability to shrug off Stupefying Charms by even several wizards at once. Constantly being around the magic of the Stupefy charm would make them immune.
Unfortunately, there is no way to answer any of these questions completely. Dragons- as well it should be- are simply too magical to be explained.
While this gives only the barest glimpse into other creature's magic, it is meant to open the reader's eyes. Simply because the being in front of you doesn't hold an Ollivander (tm) Original, does not mean it might not be able to counter your charm. Be open to magic from everywhere, and perhaps they can teach even more than a Hogwart's Teacher!