This is the question that seems to come up with anyone who isn't the most advanced of users. There are so many options, so many designs and configurations it's overwhelming to some, especially those who never knew there were any choices.
I'm going to try to clarify and simplify the selection process for you here. But first, I want to talk a little about the saw itself, since choosing the wrong saw can create real headaches when it comes to trying to buy replacement and specialty blades. Keep in mind not only the size of the saw, but the size of the arbor when selecting a new power saw. Or picking up an old one at a yard sale, estate sale, etc. There are some great old table saws out there, like the old, solid-as-a-rock Sears models that have ¾"arbor shafts, for example. Then there was the huge batch of Asian-made 10" miter saws on eBay with 1" arbors. With cheap OEM blades on them. When I saw those, I knew in a few months I'd be getting emails from folks looking for new blades for the bargain saws they bought - blades no one makes. I was right. And those bargain shoppers found out it's almost impossible to get blades off the shelf, that if they're lucky, they can get one bored to fit - at a price.
Table Saw Tools
Almost all power saws for the American retail market from about 8" to 10" come standard with a 5/8" arbor so that's what blades are made for. Anything 12" and larger comes with a 1" arbor. DeWalt makes some 12" miter saws that come with a reducer so you can use either arbor size and there are some 12" industrial double miters that (like Pistorius) that have a 5/8" shaft. But other than those, the standard arbor sizes are about written in stone so if you're shopping for a new saw, stick with them.
Okay, now about those blades...
Size (or Diameter)
This one may seem pretty obvious but once in a while I run across someone who wants to use a blade that's a different size than their saw (sometimes to save money, believe it or not). In most cases, I advise against it. First of all, there's the issue of clearance: a bigger blade won't likely clear the blade guard on a miter or radial arm or the throat plate on a table saw. A smaller blade won't give you the depth of cut.
Then there's what's not so plain to see: the design and geometry of the blade. Smaller saws run at higher RPM, the bigger the saw, the lower the RPM. Blades are designed to work in concert with the saw to give you optimal performance. Enough said about that.
Exceptions? Yes, most notably a dado. Most craftsmen will use an 8" dado on a 10" table saw and in fact, that's the only size most manufacturers make - with a 5/8" arbor bore. For the industrial market, a few companies like SystiMatic make a selection with larger sizes and 1" bore, but you'll need a pretty powerful, heavy saw to use it. A 10" SystiMatic dado weighs in at over 10 pounds and cuts a lot of material.
Another exception may be a highly specialized use, by a very knowledgeable professional, such as the one I heard about not long ago where they were using an 8" non-ferrous blade on a 10" or 12" table saw to cut aluminum plate. This was a blade designed for cutting extrusions, not solid aluminum, but these guys understand the geometry and use it to their advantage.
Purpose or Material
This one can be problematic if you're trying to buy a blade at Home Depot, Lowe's or ACE where all they sell is blades for cutting wood on hand-held saws (Skil, etc.), table saws and miter saws. But if you're shopping online or at a saw shop or good tool store, you can find specialty blades listed by purpose and material or ask someone who knows what you're talking about. And believe me, it's important to buy a blade designed for what you want to cut.
The price of a 24 tooth rip blade might look attractive but if you try to crosscut with it you won't be happy. Conversely, try to rip solid wood with a trim blade and you'll burn more wood than you cut. Use either blade to cut laminate flooring and you'll regret it when the top layer chips and the blade gets dull after three cuts. Laminate flooring, like Pergo, is very hard and has aluminum oxide in it - use an aluminum-cutting blade.
Manufacturers of saw blades that make blades for the professional make several different blades, with different configurations, for different uses and they identify them accordingly. There's no need for me to list them here, just stick with their recommendations and you shouldn't have any problems.
Type of Saw
Different types of power saws work best with blades made for them, and can work horribly (if at all) with the wrong blade. Using the example of a rip blade again: put a rip blade on a radial arm saw and even if you rotate the carriage to rip, it's going to want to lift the wood up off the table. Try to crosscut with anything vaguely resembling a rip blade (aggressive rake/hook angle) on that same saw and the whole carriage will try to "run" straight out at you. And the teeth will dig in and bind up your saw, tripping a breaker. Been there and done that one before I knew there was such a thing as a radial arm blade.
There are some types of blades, such as metal cutting blades, that can be used on and are recommended for all types of saws: table saw, miter saw and radial arm. Also some moderate-rake/hook combination blades. But a good rule of thumb is to just stick to what the manufacturer recommends. Buy a table saw blade for a table saw, a miter blade for a miter saw and a radial arm blade for a radial arm. When you think about it, you're not going to be cutting sheets of plywood on a miter or radial arm and you're probably not going to be mitering moldings on a table saw.
Tooth Count
Most people understand this one, for the most part. Generally speaking, you want a higher tooth count for cleaner, finish cuts, no matter what the material. You want fewer teeth for thicker material. Think of crown molding versus a 2 by 4. But if you're a hobbyist or homeowner doing odd jobs (like yours truly) you'll want a compromise blade, something you can use to reasonably cut a 2 by 4 or trim. On a miter saw or radial arm, 60 is a good number: 40 will tear out on trim work and 80 will have to fight through a 2 by 4. On a table saw, it's a little more forgiving: 40 to 50 is what you'll find on good combination blades, like the famous Forrest Woodworker, the TENRYU Gold Medal or the SystiMatic GP or Budke Combination. Virtually all manufacturers make combination table saw blades with tooth counts in this range so obviously it works.
By the way, as usual there are exceptions to this rule, especially when it comes to cutting plastics. And it depends on the type of plastic, whether it's hard and brittle, soft and with a low melting point or in between. Too many teeth will cause melting, which will load or gum up the teeth, thereby giving you a very poor cut. Not enough teeth, particularly in a hard plastic, will chip like crazy.
Tooth Design and Configuration
This is the one topic that can be most confusing and the one where you might, in some cases, find conflicting theories and/or claims from different manufacturers. Not to say one is right and the other is wrong, sometimes two completely different designs will work equally as well for a job. For example: SystiMatic uses a triple chip for plastic and TENRYU uses alternating top/alternating face. But I suppose I should simplify this before confusing you more.
Alternating Top Bevel - ATB This is by far the most common carbide tooth configuration, used for cutting solid wood, plywood and particle board. Further, there are several variations on the design that turn it into a specialist:
ATB w/Raker - ATBR Commonly called a "planer" blade or "planer combination" this combines usually four ATB teeth with one flat-top raker tooth for cleaning out the cut. It makes a true multi-purpose blade for your table saw, whether cutting plywood, crosscutting or ripping. A further variation on this one is a specialized plywood blade that uses more (like 10) ATB teeth for each raker.
Alternating Top Alternating Face - ATAF Very, very smooth crosscuts, with the outer edge of the tooth face chiseling the material as the blade cuts through. Also good for Melamine and veneered plywood. And as mentioned, TENRYU uses this for their plastic blades, too. Put one on a miter saw and get great cuts in wood and plastic, both. A variation on this one is to add a raker every few teeth to clean out the cut. It would be called ATAFR of course.
High or Steep Alternating Top Bevel - HATB This tooth shape, combined with a negative or neutral hook or rake angle, is used when you need a real knife-like edge to cut through Melamine or fine veneers. Also sometimes used with a positive hook for eliminating tear-out when crosscutting trim. The drawback to this type of tooth is that the quality of your cut depends on very pointed teeth and the more pointed they are, the faster they dull.
Triple Chip Grind - TCG This is a versatile tooth shape but the primary purpose is for cutting hard materials, like aluminum, laminate flooring, hardwoods and "solid surface" such as Corian. It incorporates flat top raker teeth with what look like a flat top teeth with the corners ground off at an angle. Without using sharp points like ATB or ATAF blades, a TCG blade will last much longer and handle the high impact of cutting hard stock. Manufacturers combine this shape with different hook/rake angles to specialize blades, from negative hook angles on non-ferrous blades to very aggressive hooks on rip blades. Drawback: may tend to tear out when crosscutting softer wood like pine or hemlock.
Flat Top Grind - FTG Flat top grind teeth, when used alone, have only one purpose: cutting wood with the grain. Ripping. And they've lost popularity in that use, too, as more manufacturers are using TCG and ATB teeth to give rip cuts smooth enough they don't need to be run through a jointer to glue up joints. But you've seen how other designs incorporate flat top teeth into doing their job well.
Others Some manufacturers use some highly specialized tooth shapes for their more specialized, exotic blades but we won't need to go into those here. They're far from basic and anyone who needs one of those probably knows more about it than anyone.
Rake or Hook Angles
When considering a blade for your particular type of saw, this is sometimes the most important consideration. And when combined with tooth shapes and configurations, the factor of rake angle can change a blade's entire purpose.
Where the saw is concerned, you don't want to use a positive hook on a radial arm (some manufacturers rate their combination blades for them but they're not aggressive hooks) and a miter saw works best with negative, neutral or moderately positive hooks - depending on what you're cutting.
Where material is concerned, generally harder materials require a negative hook or no (neutral) hook angle. And if wood is prone to tearing out when crosscutting, like softer conifer wood, a negative hook is better. So what is rake or hook angle? Lay a straight edge across a saw blade, intersecting the arbor hole and look at the relationship of the carbide tip to the straight edge. If the top of the tip leans toward the edge, that's a positive hook or rake. If it leans back away from it, that's negative. If it's parallel to it that's neutral or a zero hook.
I hope this clears up some confusion about selecting the right blade for the right machine, material and the right job. You want the best results, you need the right tool.