Adam, the bobbin numbers are from the old Singer line. Singer was the industry standard so everyone imitated them. The class 15 bobbins were used in the 15 series machines and are probably the most common bobbins out there. My Bernina, albeit an old one, will accept a 15 bobbin in a pinch and my new Brother Inovis 40 uses a class 15 bobbin. I could not get the saleslady to tell me exactly what size bobbin I needed with the Brother (they want you to buy their brand specific ones) but I have been using my Singer 15s in it for two years now. I have had other machines that used the 15 bobbin as well. The 66 bobbin goes in a Singer 201 and several other Singers. It is smaller than a 15 (holds less thread) and is a drop in, at least on my 201. The 66 bobbin is almost as common as the 15.
Can't say anything about bobbins and newer machines, other than my Brother, because I have no experience with them. Don't toss what you were given. If you are a machinist you will have the measuring tools to just match them up by size to a known 15 or 66. If you have any that don't match up you may want to pitch those or pass them on down the line. Personally, I prefer the old metal bobins to the plastic, but that is me. Jenny, over at Sew Classic may have more information.