There are two factors that go into grading eggs: their size (Pee Wee through Jumbo), and their quality (C through AA). Just as an example, the last eggs I bought at the store were Grade AA Large.
The weight of the eggs is pretty straightforward. I'm a knitter, so I have a small digital scale which I use to weigh out sock yarn by the gram. A small kitchen scale may give you enough resolution to weigh your eggs properly. You can also buy egg scales, which is probably only useful if you plan to sell your eggs.
Jumbo - 2.5 oz, or 70 grams
Extra Large - 2.2 oz, or 64 grams
Large - 2 oz, or 56 grams
Medium - 1.7 oz, or 49 grams
Small - 1.5 oz, or 42 grams
Pee Wee - 1.2 oz, less than 42 grams
The quality of the egg runs from C through AA. Two things affect an egg's grade: the quality of the yolk and white, and the presence or absence of certain flaws which don't affect the taste of the egg.
For example, a small blood spot or an off-center yolk will drop an egg to Grade C. An egg that is otherwise fine, but has an oddly shaped shell, would probably be considered Grade B. Obviously, these are only considerations if you plan to sell your eggs. And even so, my experience is that most people selling their own eggs are willing to overlook some cosmetic flaws - as are their customers! After all, an oddly-shaped egg tastes just as good as a perfectly egg-shaped egg.
I can't imagine you would have a problem with the quality of your egg yolks and whites, since this is largely determined by freshness and by the chicken's health and nutrition. However, you may get some strange, poor eggs when your chicken starts laying, including eggs without yolks, or eggs where the white is unusually runny. However, once your chicken is up and running (so to speak) these quality issues should level right off.
Most of the quality flaws in eggs result from their age - as an egg ages, the air cell gets larger, the yolk loses its springiness, and the egg white starts to turn runny. I've read that grocery store eggs can be up to a month old by the time you buy them! And a while back, there was a minor scandal about eggs being repackaged after a month to "reset the clock," which meant that the eggs could be six months old by the time someone bought them. Yum!
Most of us have only encountered Grade A and Grade AA eggs, since this is what is sold at the grocery stores. Grade B eggs are usually sold directly to restaurants, hospitals, and large cafeterias, since they are most suited to being scrambled or used in baking. Grade C eggs are so poor and runny that they are typically shipped directly off to processors which use the eggs as an ingredient.