The general topic you are referring to is called operant conditioning.
Positive/negative refers to whether you are adding something or taking it away.
Reinforcement/punishment refers to whether you are causing a behavior to increase or decrease.
Positive punishment is adding (positive) a bad thing (punishment) like yelling at or slapping the kid.
Negative reinforcement is taking away (negative) a bad thing (reinforcing).
Negative punishment is taking away (negative) a good thing (punishing), like limiting access to a certain toy or the television.
Both positive punishment and negative punishment tend to reduce a behavior, though often positive punishment is less effective and more likely to produce off-effects.
Importantly, in operant conditioning everything is relative to the consequences of a behavior by the one behaving ("operating"). If someone is not "operating" you cannot describe it in operant conditioning terms.
Your example scenario
Your first example is positive reinforcement from the perspective of the child: their behavior (begging) leads to getting rewarded with lollies. One would expect that this would increase future begging for candy rather than reduce it.
The only way to think of this as negative reinforcement is from the perspective of the parent, for whom giving the child lollies (the parent's behavior) has removed a noxious stimulus (the begging by the child), which might lead the parent to again purchase lollies to reduce complaining.
However, in parenting or training a dog or most any other context where there is a power relationship, you are usually more concerned with thinking about how behaviors affect the child/dog/whomever. In this situation, if a parent wants to stop the begging long term, likely the best approach is to target extinction of the behavior, which occurs when a behavior (e.g., begging) does not achieve a positive outcome (e.g., no lollies are obtained).
When you write:
I would've thought that the if the son's behaviour changed from misbehaving, this is an example of a punishment (albeit a tasty punishment)
From the perspective of the son, this is not an example of operant conditioning, because the behaviour change (stopping begging) is not causing the 'stimulus' (getting candy), but rather the other way around. You could only consider this operant conditioning from the perspective of the parent, who is being trained to do an action ("give candy") to remove an aversive stimulus (child begging).