Grades can be seen when you click on your monster and press ''Identify'' (it costs a mere 100 Silver).
If you look at the picture below, you'll see that the monster has an avarage grade of D- ( Next to the EXP ), and several individual stats in D+, D- and even an E stat.

Now, if we look at another monster, you'll see that its avarage grade is S ( Again, next to the EXP ).
All other stats are S aswell, besides DEF, which is A+.

Simply put : If you have two of the same monster that are both Level 60, but one is D- grade and the other is S grade, the S grade will have better stats.
As said before, the only way to increase the Grade is via Breeding, which requires the Hatchery, being Level. 15, a
male and female parent, some Silver ( Ranging from 5K to 58K ) and Love Potions.