This page outlines the basics of Motivation & Bothering.
Motivation - how does it work?Edit
When you start a Work, Errand, or Study, you'll notice that your fairy has something called Motivation. Your fairy can have from 1 to 200 motivation, but it usually starts at 100. If your fairy has a lower mood or affection than the maximum, it may start with less motivation. (The starting Motivation of fairies "working together" depends on their intimacy with each other.)
Motivation modifies how many stats, jewels, and exp your fairy receives when it is done with its task. A fairy finishing its task with 200 motivation receives double the rewards than a fairy finishing with 100 motivation. (And, conversely, a fairy ending a task with less than 100 motivation receives far fewer rewards!) Thus, if you want to get the most benefit out of your task, you will want to raise your fairy's motivation! This leads us to...
Bothering - what is it, and should I do it?Edit
You may have noticed that you can Bother other people's fairies while they are doing a task. That sounds troublesome - should you do it? Yes! Bothering puts a person's fairy in a distracted state and lowers its motivation by 10 points. However, it allows the fairy's owner to then yell at the fairy to motivate it, raising motivation by 20 points and putting it into an excited state for 10 minutes, during which no one can bother it again. This means that an owner who remembers to yell at their family gets a net gain of +10 points motivation.
One thing to keep in mind is that a fairy ending its task in a distracted state slowly loses affection and mood. Therefore, it's a good idea not to bother an owner's fairy if there is a short period of time left on the task (when is a bad time varies by owner, but if there are 5-10 minutes left it is probably a good idea to leave them alone unless you know they are online. note: as the fairy's owner, it's a good idea to try to cover your last 10 minutes with an 'excited' state to prevent this very thing from happening!).
You should also not bother a fairy when it is at 200 motivation, since fairies can't go beyond this motivation and putting it at a distracted state would cause no net gain to the owner.