There is a bit of a disagreement among mythologists as to what really constitutes a werewolf. I'll break it down into the three distinct creatures that are all known by this name.
1. The Shapeshifter Wolf
Fans of the Twilight series will recognize this type of "werewolf". Like the character Jacob Black and certain other members of the Quileute tribe, the shapeshifter wolf has the ability to change form at any time. This is believed to have been the original power of the first werewolf before it was taken away. The shapeshifter wolf can transform from human form to wolf form at will, though aggressive energy or anger may increase the likelihood of causing an unintentional transformation. The shapeshifter wolves considered "werewolves" can only change from human to wolf form, though "pure" shapeshifters can transform from human form to any other animal form, including a wolf. It is believed that all shapeshifters are born into this ability, and one cannot be transformed into a shapeshifter by the bite from the creature. The exception to this is the Navajo Skinwalker, which is not a true shapeshifter, but a witch using an animal hide to transform.
2. The Wolfman
Many "werewolf" stories, including sightings of the legendary Bray Road Beast, describe a creature that is physically a combination of a wolf and a man. This mutant wolf-man typically has a mostly human-shaped body, stands on two legs, but is covered in wolf hair and has claws and fangs. The Wolfman is often called a "werewolf" because in many legends this creature is believed to share many of the traits of the werewolf "curse". Depending on the legend, some wolfmen are humans by day and wolf-men by night while others are in their duel human/wolf form at all times. I believe the generally accepted theory is that one begins by transforming from human to wolfman on full moons only, then over time it increases to every night, and eventually the body settles into a pure wolfman form at all times. The degree to which a wolfman can control his mind and emotions also degrades as these transformations become more permanent.
3. The True Werewolf
A true "werewolf", according to most legends, is a human being that uncontrollably transforms into a wolf during a full moon. The original werewolf curse transformed a Shapeshifter Wolf into a Werewolf, effectively taking away the ability to control the shift but also taking away the ability to think as a human while in wolf form. Werewolves, when in wolf form, have uncontrollable rage and hunger. They are driven to kill everyone and everything they encounter, regardless of their relationship as humans. The curse causes these wolves to lose all control of their minds, and when they wake up in human form in the morning, they don't remember anything (though they may revisit certain memories in dreams). It is this form of werewolf that can transfer their condition through a bite, assuming of course that the human being bitten survives the attack.