Understanding the structure of the natural nail is important for a number of reasons. One, you want to know what you are working with. Two, you want to be able to talk professionally about your job. Three, many professional nail companies use proper terminology and you want to be able to understand what they are talking about.
One of the most misunderstood aspects of nail structure is the cuticle - I wrote a post comparing cuticle/eponychium /pterygium
Personally I find the mechanism of nail growth fascinating - you can see in the picture below that the nail grows up from the matrix and as it is pushed out onto the nail plate, it collects dead skin cells from the eponychium - this is what is called true cuticle. The cells that are produced in the matrix are soft and round and as the gather and start being pushed up to the nail bed they are flattened into layers and become harder. One interesting bit of trivia is that the lunula (the white "moon" at the base of some people's nails) is actually the top edge of the matrix. The size of your lunula is due to the position of your matrix on your finger - absolutely nothing can change that (I have heard of people saying "just push back your cuticles"....yeah...no.)
You may ask yourself - how does the nail plate stay on the nail bed? Take a very, very close look at your nail bed (assuming you don't have polish on!) and you will see little "lines" that run the length of the nail - also seen in the third picture below. These lines are like little rails in the nail plate that fit into grooves on the nail bed. The thinner your nail the more likely you are to feel these lines in your nail. The hyponychium under your natural nail free edge "seals" the nail plate to the nail bed. Too aggressive of cleaning under the nail can break this seal and then the nail can start to come up off of the nail bed on these rails. And when you hit your nail with a hammer and get little "splintery" things? Nope, not splinters - these are called splinter hemorrhages - they are caused by trauma - less trauma than what would cause a blackened, bruised nail, but enough that a little bit of blood gathers in these nail bed grooves. You just need to let them grow out, there is nothing you can do for them otherwise.
I am not going to go super-in-depth about the rest of the structure of the nail as this is something you should have learned in school. BUT if you have questions please feel free to post them!
I encourage everyone to become familiar with the PROPER terminology and use it!