Infantile means immature or childish. Thumb-sucking and temper tantrums are examples of infantile behavior.
Doctors use the word infantile just like you'd expect: to describe things having to do with infants or babies. The rest of us, though, generally use this adjective when someone's being a brat or a crybaby or otherwise behaving immaturely. If someone refers to your sense of humor or your style of argument as "infantile" they are not offering you a complement. They might be trying to tell you to "grow up."