What do the Pokemon "Characteristics" mean?

It shows the highest IV, and its value modulo 5 (check out How do IV stats work? for information about IVs). If there are mutliple highest IVs, the value is choosed randomly depending on personality value (which is also random). It can be used to easily determine which IV is highest.

Depending on value of IV, the following characteristics appear. For example, "Takes plenty of siestas" could mean HP IV ending with 1 or 6 (but having just 1 HP IV is rather unlikely, as it shows the highest IV, and that would mean other statistics are even lower - but unless you are abusing RNG or are just that unlucky, the only situation where that actually could happen is Roaming IV Glitch in third generation).

• The highest IV is HP
  • 0, 5, 10, 15, 20, 25, 30: Loves to eat
  • 1, 6, 11, 16, 21, 26, 31: Takes plenty of siestas (Often dozes off before Gen6)
  • 2, 7, 12, 17, 22, 27: Nods off a lot (Often scatters things before Gen6)
  • 3, 8, 13, 18, 23, 28: Scatters things often
  • 4, 9, 14, 19, 24, 29: Likes to relax
• The highest IV is Attack
  • 0, 5, 10, 15, 20, 25, 30: Proud of its power
  • 1, 6, 11, 16, 21, 26, 31: Likes to thrash about
  • 2, 7, 12, 17, 22, 27: A little quick tempered
  • 3, 8, 13, 18, 23, 28: Likes to fight
  • 4, 9, 14, 19, 24, 29: Quick tempered
• The highest IV is Defense
  • 0, 5, 10, 15, 20, 25, 30: Sturdy body
  • 1, 6, 11, 16, 21, 26, 31: Capable of taking hits
  • 2, 7, 12, 17, 22, 27: Highly persistent
  • 3, 8, 13, 18, 23, 28: Good endurance
  • 4, 9, 14, 19, 24, 29: Good perseverance
• The highest IV is Special Attack
  • 0, 5, 10, 15, 20, 25, 30: Highly curious
  • 1, 6, 11, 16, 21, 26, 31: Mischievous
  • 2, 7, 12, 17, 22, 27: Thoroughly cunning
  • 3, 8, 13, 18, 23, 28: Often lost in thought
  • 4, 9, 14, 19, 24, 29: Very finicky
• The highest IV is Special Defense
  • 0, 5, 10, 15, 20, 25, 30: Strong willed
  • 1, 6, 11, 16, 21, 26, 31: Somewhat vain
  • 2, 7, 12, 17, 22, 27: Strongly defiant
  • 3, 8, 13, 18, 23, 28: Hates to lose
  • 4, 9, 14, 19, 24, 29: Somewhat stubborn
• The highest IV is Speed
  • 0, 5, 10, 15, 20, 25, 30: Likes to run
  • 1, 6, 11, 16, 21, 26, 31: Alert to sounds
  • 2, 7, 12, 17, 22, 27: Impetuous and silly
  • 3, 8, 13, 18, 23, 28: Somewhat of a clown
  • 4, 9, 14, 19, 24, 29: Quick to flee