Man, I wanted to answer this last night, but as has been mentioned earlier, it's a large can'o worms. Doing so on a phone wasn't ideal. I'll try and keep it succinct, I already know this is going to be a long post, so appologies beforehand. This is all based on my experience, so take it with a grain of salt as your experience may differ!
So, just speaking badger brushes, Dave was pretty spot on with his list of badger types. However, some manufacturers will take some liberty with what they call certain grades. Each grade is established based on how the knot prepared and set in glue, generally. Properties and terms generally used to describe these are loft, backbone/floppiness, scratchiness, scritch, scrub, density, presence, flow through, gel-tips, fan/bulb shape, and 2-band/3-band.
Loft - We'll talk about free loft here, which is the distance between the base of the handle where the knot enters, and the tips of the hair. A good frame of reference is the loft should be about 2x the diameter of the knot. So, a 24mm knot would have a loft around 48mm.
Backbone - this is typically determined by the shaft thickness of the hair, and can be described as how difficult it is to bend the hairs. A brush with good backbone is typically better for loading soaps and providing good exfoliation on the skin due to the increased pressure used to splay the hairs out. Outside of hair type, the backbone can be increased by setting the knot deeper into the handle (decreasing the loft), or higher (increasing loft).
Scratch / Scritch / Scrub - About the most subjective terms in the shaving world, since they mean different things to everyone. For ME, a scratchy brush if used to face lather can feel like steel wool after too much face lathering. Individual hairs can be felt. Some people like this property, though I'm not a huge fan. Scritch and scrub, for me at least, are almost interchangeable. Scritch is almost the same as scratchy, but softer feeling on the face. Combined with good backbone gives you a scrubby brush; for me the most preferable.
Density / Presence - density is the amount of individual hair that is packed into a knot. A brush that is ultra-dense can be what's known as a "lather hog" and be difficult to get lather out of the brush. A brush with good density will retain heat very well. Presence is a term that is used by Ken Lynn of Paladin to describe the combination of Density, Backbone, and Scrubbiness...and I think it's a great term to describe brushes that combine those qualities well.
Flow Through - This is best described as how well the brush releases the lather from the knot. Some of the most luxurious feeling and dense knots (Simpson Chubby 3 in Super for example), do not typically have good flow through and are called lather hogs. Some don't mind this, others can't stand it. It's all about finding what you prefer.
Gel-Tips - You'll see, on occasion, people say that the tips of their brush were gel-like. This is typically used in reference to High Mountain White or high grade Silvertip brushes which the tips are so fine that when wet (not loaded with lather) they clump together and the feeling is just like the term sounds...like gel.
2-band / 3-band - This is kind of a misnomer, but it refers to the visual look of the brush when it's set into a handle. The banding is the change in coloration from light to dark (2-band) and back to light (3-band). I say it's a misnomer as if you saw the knots out of the handle, most would actually have 3-bands...you just can't see the 3rd band as the dark 2nd band is much wider, hiding the lighter band inside the handle. 2-band brushes typically display more backbone than 3-banded brushes.
Fan / Bulb - Just refers to the shape of the knot at the tips. They have slightly different properties as far as feel is concerned. At the same loft, (in my experience), a fan will not have quite as much backbone/presence as a bulb will. Some say one is more conducive to bowl lathering or face lathering, but I use both for either.
Badger / Black Badger / Pure Badger / Mixed Badger - Generally are uniform in the shaft thickness along the entire length of the hair, and the tips are actually cut to shape once glued into a knot. They tend to be more scratchy since the tips are cut, less dense (amount of hair packed into the knot), and from an aesthetic standpoint, will be brown or black in coloration with no "banding". They aren't bad brushes, as long as they aren't shedders (where the hair breaks out of the glue holding them in), but they aren't luxury brushes either. They don't retain heat for very long, due to the lack of density, either.
Best Badger - This is a grade that is used by Simpson which is their lower/mid grade, but is much more luxurious feeling than the above referenced knot types, as the tips are not cut to shape and have their natural ends on them. Typically 3-banded, they aren't quite as dense as their Super Badger cousins, with tips that are a bit more scratchy/scritchy.
The following are all typically considered "Silvertip" badger brushes...referencing the light colored tips of the brush.
Finest Badger - typically 2-banded, the shafts of each hair is a bit thicker than the above hair types, so display a greater amount of backbone, but unlike the previously discussed knots, the hairs narrow out to a pretty fine tip. There's a pretty wide variance in quality here, so density of the knots, hair quality, shaft thickness all vary. This is my personal favorite of the silvertip grade knots. IMHO you get the best bang for your buck in regards to knot quality and face feel.
Silvertip / Super Silvertip 3-band - Generally these have less backbone than both Finest and High Mountain White but with much softer/finer tips. Considered by many to be the most luxurious feeling on the face, the tendency is to be a bit more on the floppy side of the house due to the decrease in backbone. This is most likely due to the shaft being much more thin than its 2-band brethren along the length. They can make up for this in part by sometimes being much more densely packed, or decreasing the loft.
High Mountain White / Manchurian - Can be 2 or 3 banded, typically with much more backbone, and higher density. Manchurian's are most often 2-banded, but as in my Brad Sears brush, can have 3-bands. "Manchurian" as a term does not reference that the badgers came from Manchuria, but started out as more of a marketing tool used by Simpson for their 2-band silvertip knots. I tend to think of these as very high-grade Finest knots.
HMW and Silvertip graded hair is generally a reference to where on the badger the hair comes from, which is a very specific, and relatively small, portion of the badger's hide. Because of this, they command a higher price. There are many different brush and knot makers who use some of these terms interchangeably...so it can get a little confusing.
A ton of information, but hopefully it's helpful to you. As with anything else in this hobby, if what you have is something you enjoy, then that is what's best for you.
