It is fairly obvious that Witches and Wizards are possessed of some trait (Muggles may call them "genes") that allow them to use magic and interact with the Wizarding World in ways that Muggles cannot. Many different animals, plants and fungi also have magical properties. That is, they have either extraordinary abilities (in the case of magical creatures) or extraordinary properties (in the case of magical plants and fungi). 

The most obvious thing that Witches and Wizards can do but Muggles cannot is use magic. Muggles can neither cast spells nor sense the presence of magic. They also are unable to brew potions, even if they happen to have the proper magical ingredients, due to the need to infuse a potion with magical power while the potion is being brewed. Obvious active powers (such as spellcasting) aren't the only thing magical folk have going for them. There are also several passive abilities that a Witch or Wizard possesses that Muggles lack.

A good example of this lies with the creature known as the Dementor. Dementors are responsible for guarding the Wizard Prison Azkaban, and one of their magical talents is the draining of happiness from their surroundings. Witches and Wizards are able to see a Dementor, though Muggles and the non-magical but wizard-born Squibs cannot*. The only thing a Muggle can sense is the happiness being drained from them, and when they are incapable of understanding why this is happening, the experience is even more frightening.

Witches and Wizards can also inherently use devices such as Broomsticks while Muggles cannot. Such things aren't necessarily powered by magical energy, but instead may require only the presence or intermingling of magical power. Some devices are magically bewitched in such a way that being magical is not necessary to "use" them, though in these cases the items are typically prank items illegally enchanted and used against Muggles. Examples include vicious toilets and biting teakettles among others.

The process by which one gains magical ability is well known. In most cases, the child of a Witch or Wizard will themselves have magical ability. Rarely the offspring of a Witch and Wizard can be a Squib, someone who is not able to use magic nor use magical devices, such as brooms, that require the user to have magic. Occasionally it's possible that two Muggles are able to have a magical child. There is always a Witch or Wizard somewhere within their family history, though they might be several generations removed. Being the offspring of two Muggles or having at least one magical parent does not appear to affect a Witch or Wizard's aptitude for magic.

The offspring of two magical parents who have never intermixed with Muggles is often referred to as "Pureblood". Those who are known to have at least one Muggle parent, grandparent, etc. in recent memory (as well as at least one Witch or Wizard) are known as "Halfbloods". For those who have no documented Witches or Wizards in their ancestry, the term "Muggleborn" (and occasionally the highly offensive term "Mudblood") are used, though the offensive representation is considered a fairly egregious insult.

A Witch or Wizard has magic coursing through their veins in the form of magical blood, hence why there are categories of blood status. The source of the magic is unknown, though popular theories speculate that it may come from some unique way a magical being processes what they eat or drink, or perhaps even the air that they breathe. The fact that it can exist in so many creatures and organisms and yet still elude Muggles is a fascinating discussion best saved for another time.

Witches and Wizards are able to manipulate their environment without the use of a wand, though this is typically the result of a young magical child unable to control their magic in times of emotional stress. Some find that they can continue to use such talent, even after they have control over their magic, but most don't find the need. Young students quickly learn how to prevent magical leakage as well as to direct their power out of themselves, through their wand, when casting spells. Of course, a few highly accomplished witches and wizards can cast spells without a wand, but this is very rare. The next chapter deals with the components of casting such spells.

Squibs: Almost the opposite of a Muggle-born wizard: a Squib is a non-magical person born to known magical parents. Squibs are rare; magic is a dominant and resilient gene. 

*(It has been established by Rowling that Arabella Figg was lying about what she said in Harry Potter's defense. Squibs know what a Dementor is and what it feels like, but they cannot see them.)