What do 'hitbox' and 'hurtbox' mean?

In most games Hitboxes and Hurtboxes are the same thing. They both refer to the area on the entity in which you (or more specifically, your own hitbox/hurtbox) can come into contact with for selection (talk to, or to interact with), or dealing damage.

As an example, Minecraft hitboxes are indicated by the white wire-frames surrounding the mobs, and items. (The hitbox representation for the player is glitched in this image - caused by the 3rd person camera offset):

Technically, this is the same "area" that can also deal damage to you (when a player punches another player, or swings at them with a weapon), it is based on this same area. an arrow, on the other hand, is dependant on its "state" - if it has been fired from a bow, the "hitbox" of the arrow turns into a "hurtbox" - anything the can receive damage from it will take damage if it hits. Otherwise, if it is not in flight (sitting on the ground, or stuck in a wall to be picked up), it turns back into a "hitbox".

As you can see: interacting/swinging at the pig's head will yield nothing, as it is not within the bounds of the hitbox.

However, in other games, mainly fighting games, like Street Fighter or Tekken, they are different. The hitbox is the area that will hit the opponent (red area), while the hurtbox is the area that you can take damage from an enemy attack during your attack (green box):

In addition to this, there are also some games that can have multiple versions of hitboxes on the same entity. For example, in games like CoD, the head has a different hitbox to the body, and limbs, as hitting an enemy in each specific area would do more or less damage (I.e. a "headshot" would do the most, body less, and the limbs would do the least amount of damage per "hit").

In fighting games, in particular, a hitbox is the area of the screen that will do damage if it touches an enemy hurtbox. A hurtbox is the area of the screen where a character is vulnerable to attack.

In this image, the blue box is the hurtbox, while the red is the hitbox.

In other genres, hitbox is often used to refer to both the area of an attack, and the outline of a character used in hit detection.