Card stocks vary, generally based on thickness and pliability. Card finishes are a separate feature of a playing card - it's what they've been coated with and whether a texture has been applied to it. The process is usually like this:
Card stock paper arrives at the card company on heavy rolls, straight from the mill. Two rolls are glued and pressed to each other to make a single, bigger roll of thicker paper - what we call pasteboard. The glue has graphite mixed in to make the playing cards opaque when held up to the light.
Depending on the volume and type of cards being made, these two steps occur but sometimes in reversed order. In other words:
Either the card paper is cut into sheets and the sheets are fed into a "sheet press" printer, as used for relatively small deck print runs, or...
The paper in roll form is fed into a "web press" printer, printed, then cut into sheets, a process usually reserved for LARGE print runs of high quality, such as premium casino-grade playing cards.
People have argued which method is better, with people often declaring that "web press" is superior to "sheet press". Well, the current state-of-the-art playing card finish, known by USPC as "Magic Finish" but also referred to by some playing card companies as "Performance Coating" or "Premium Finish", is only available on sheet press runs.
During the printing process, after the ink has hit the paper, the cards are coated with a finish, usually consisting of some plastic-type chemical laminates. At this point, the cards finish may also be either left smooth, or given a proprietary patterning. The chemical and pattern combined form what's known as the finish, and there are many out there, most of which are considered trademarks, such as "Air Cushion Finish" on Bicycles, "Linoid Finish" on Tally-Ho decks, "Cambric" (textured) and "Ivory" (smooth) finishes on Bees, "Magic Finish" (see above) on many privately-produced decks, etc. "Smooth" is a generic finish term - while most smooth-finish decks are of a lower grade, a major exception would be the Aladdin deck in the smooth finish - a high-grade deck designed for use in hot, humid climates, considered to have exceptional fanning qualities for a smooth card. "Linen Finish" is still used as a term, but a true linen finish hasn't been used in perhaps thirty years or so - the rollers used to create the linen texturing on the surface of the card used to be made of different linens and cottons, but are now almost exclusively made out of steel.
From here, the card sheets are cut - first into strips, then the strips are cut into the individual cards. Modern plants do the cutting and sorting by machine, while older facilities use manual cutting and sorting methods. After the cards are cut out of the sheets, they're still rectangular - the corners are rounded by putting them into a die press; usually a single die used for each card in the deck to insure that the entire deck has the exact same shape for each card. If a card is cut "face up", it's fed through the die so the back of the card is penetrated first - this is the modern method for cutting, supposedly more efficient, but it also results in cards that don't have superior shuffling unless they're shuffled face up. A "face down" cut, where the card is cut with the die piercing from the front first through to the back, is often called a "traditional" cut, since up until the 1980s all decks were being cut this way. Casinos to this day usually insist on getting traditionally-cut decks, since they have superior shuffling when face down, especially when performing weave and faro shuffles. Many magicians prefer this type of cut, and certain decks designed specifically for magicians are traditionally cut, such as the Richard Turner Gold Seal Bicycle Rider Backs, the Bicycle Mandolin Backs, the Bee decks designed for the CARC, etc. Of all standard non-specialty decks on the market made by the USPC, Tally Ho is the only one that's still traditionally-cut to this day.
Now, there's all kinds of factors that makes any one deck superior to another, and that's strictly speaking in terms of paper and finish, not card design.
Certain papers are just plain cheap and inferior. USPC doesn't deal with them much. Cheap manufacturers in Asia will often make decks inexpensively out of cheaper paper and finishes, usually for decks meant more as souvenirs than as something meant for serious use. Sometimes, unscrupulous manufacturers will counterfeit other companies' designs using inferior material and processes as a way to make a buck of someone else's intellectual property.
Now, discounting those completely, the papers you'll run into more commonly will start with simple smooth-finish decks like Congress bridge cards. They aren't fancy, but they're pretty and they get the job done for a few rounds of bridge with the neighbors. If you're into that sort of thing!
Of the specialty card market, the low end of the grade list would be basic Bicycle stock, sometimes called by its stock number, 325. Not the worst, not the best - a decent workmanlike stock. I could be mistaken, but your Sentinels you mentioned may be a grade along these lines, or slightly better. They are rather thin compared to many other custom decks on the market, but some find this desirable - you can more easily create a crimp in a card that's thin, a technique used for locating a card by some magicians and card sharps. The crimp may be harder to remove, however, with a grater chance of becoming a crease - a drawback.
As you step up, there are a variety of grades which people rate in different ways. The Monarchs, which you mentioned in particular, are a variant of the Aristocrat stock, which is considered one of the best on the market - soft, pliable, but still sturdy; an excellent card player's stock. Tally Ho stock has a certain firmness and ability to bounce back from crimping (intentional and accidental) that's often preferred by cardists - people who perform cardistry, or playing card manipulation, as an art form in itself rather than as a subset of the art form of playing card magic and sleight-of-hand. Many think of Bee casino grade stock as the top for its legendary durability.
Ellusionist used to have a signature card stock and finish combination called "UV500 Air-Flow Finish", with UV500 being the stock used. In addition to being a thicker stock than most and much more durable, they also had the unique property of glowing under an ultraviolet or "black" light. But due to dramatically increased costs, the UV500 stock was discontinued, and USPC Legal declared around the same time that any Bicycle-branded deck that mentioned a finish had to be labeled "Air Cushion" regardless of the actual finish being used. This restriction seems to have been relaxed, however, with the release of Magic Finish, now used by the majority of custom playing card producers who want to make a top-grade deck. Ellusionist does still use thicker stocks in some older deck designs, but many newer ones aren't as thick.
Also, David Blaine just released a new blue White Lions Series A deck in the "red" or "rainbow" box that uses a UV stock, which could mean the UV500 or something similar is being revived. The deck in question was only sold as part of his "Variety Box", released last week, and only 250 of these decks are in circulation at this time. The way to tell them apart from the red and purple decks released in the same tuckbox is by looking for a small black dot in magic marker written on the cellophane so as to appear on the deck seal.
Are any of these stocks "the best" above all else? What about the finishes used on them? That depends on the criteria you choose to apply. It's like declaring one specific work of art as the world's best work of art - it can't be done because there are so many factors involved in the comparison. The best you'd be able to come up with is the best in a single person's opinion, or a small group's opinion.
How do you know which stock and finish combination are right for you? Well, try them! Sure, it's expensive to buy all the different combinations, but you can also get advice from similarly-minded collectors, players, artists, etc. who like what you like.
There are two things that most card people will agree on.
1) Magic Finish is indeed the slickest finish you're likely to find for the indefinite future, and highly desired in most newly-made decks.
2) Older playing card stock is more desirable to newer stocks, due to the older stocks being made of more original wood pulp from higher grade woods, while newer stocks have more and more recycled content with each passing year, decreasing the original wood pulp content and the quality of the paper itself. Recycling is a good thing for the environment, but a terrible thing for creating a quality paper product.
I hope this answers any questions you have right now - and steers you to better questions for the future!
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?? I probably sound really annoying to both of you, but oh well I'll try to be better ok
