Grades are a measure of how much contrast there will be in the print. The more contrast the more pure black and pure white there will be with just a few greys in the middle. The less contrast there is then the more of the picture is just shades of grey with, perhaps, nothing truly pure black or white. Lots of contrast is known as Hard, less contrast is known as Soft. In number terms Hard/Contrasty is Grade 5, Soft/low contrast is Grade 0.
In the old days all paper came in a specific grade. You could buy grade 1 or grade 4 or, as you have a box of, grade 3. Nowadays most paper is Multidgrade, that is it can be made to behave like any grade between Grade 0 and Grade 5 by using the correct filter. (Filters don't have any effect on Fixed Grade Paper).
There are two reasons for having different grades. On one hand some people prefer to have contrasty prints, some prefer soft prints. So obviously they would print on different grades.
Then there are the different negatives. Depending on the light when you took the picture and how you processed your negatives some negatives will be contrasty, some less so. If you wish to have the final prints look the same then you need to print the contrasty negative on a less contrasty/softer/lower grade paper and the less contrasty negative on a harder/higher grade number paper.
Some people (Ansel Adams for example) was very meticulous about adjusting his exposures and developing times for each and every picture he took (on sheet film) so that it would print just perfectly on Grade 2 paper. With roll film and changing light this just doesn't work so well so you'll always have some variations in your negatives, and hence the need for different grades of paper to make consistent looking prints. Hence the popularity of Multigrade paper, where you can get all those grades in just one box.
The best way to get a feel for the different grades is to use some of your Multigrade paper and print a picture at different grades. Do one at the same time for Grades 1, 2 and 3 and at double that time for Grade 4 (always double the time for Grades 4 and 5, don't ask me why). You'll see the change. There's no way to explain it in a way you'll understand it in words.
That all said my first 100 sheets of paper were all fixed grade, grade 2 in my case, and practicing on that taught me an awful lot about burning and dodging, so get going on your Grade 3 and see what you can produce and then open up the Multigrade and see how much easier it is with different Grades available. Unfortunately I am sort of with everyone else in thinking that you are going to need more paper, testing and trialling and investigating the variations being the key to understanding the differences and learning how to deal with them. And when learning about Grades perhaps cut the 8x10 paper in half to make it go further.
Most of all have fun, and good luck,
Hywel