Loyalty (or Trust) is a statistic that didn't factor into the earlier games as much. It is often not listed with the other statistics on the monster's diagnostic page, but loyalty is still very important. When you first unlock a monster, its loyalty will be 0 because it does not know you yet. If you treat your monster well and win tournaments with it, its loyalty will increase. Until then, it may be hard to make it follow your commands in battle, and often when you try to feed it or make it rest, it may second-guess your commands and gain a negative attribute called Willful.
If you treat your monster badly (i.e., battle with it all the time without rest, don't feed it right or constantly scold it for not doing well in training) its loyalty may drop and it may eventually run away. It will take quite a bit of time and money to get it back, so be careful about this. It is a fine balance between giving in to what it wants and doing what is best for it.
There are ancient legends throughout the world referring to a forbidden bond between humans and monsters. Sadly, in most cases, no good came of this bond. The most common story tells of people being tricked by a Pixie out of their entire fortune. Breeders should always take care when dealing with such monsters.
-"Monsters' Love," MR4