IS mounts (also frequently called IS tabs, disc tabs, etc) are unthreaded eyelets 51mm apart that bolts run through parallel with the hub axle. In almost all cases the brake caliper is then bolted to an adapter that has one set of threads for the IS mounts and another 74mm apart for the caliper. (There are a very few caliper models that omit the adapter, namely XTR BR-M975.)
IS has one clear upside: all the threads are in a relatively cheap and easy to replace adapter. The disadvantages are that it's more mechanically complex and expensive since there's always an adapter, and the design of the standard is such that the same adapter doesn't adapt the same caliper to the same rotor size front and rear. A adapter for 180mm in the front is an adapter for 160 used in the rear, for example.
With IS, outside of a weird custom fork, 160mm is the smallest size you can use in front, but all bikes with rear IS mounts can take a 140mm adapter.
Post mount places two threaded M6x1 holes 74mm apart on the frame or fork in the orientation that a caliper can be bolted to directly without any adapter, or with a post-mount adapter in between along with its supplied long bolts if a larger rotor is going to be used. The most common setup is for the minimum rotor size (the size you'd get the caliper positioned for if no adapter was used) to be 160 front/160 rear on most mountain bikes and 160 front/140 rear on most road and cross bikes, but all sorts of exceptions exist, i.e. downhill forks with 200mm minimum rotor sizes. Although it's seldom denoted this way by frame manufacturers, post mount is really a group of standards that you could call "140mm post mount", "160mm post mount", etc.
Post mount is mechanically simpler, cheaper, and arguably more elegant. It has the disadvantage of putting the threads in the frame/fork. How big of a deal this is is a fairly contentious subject. Personally I'm not all that troubled by it, but it does bite people. Note that it's totally acceptable to helicoil a stripped post mount in most circumstances, so the actual risk of ruining your bike forever with one overzealous tightening is pretty small. I have seen new bikes come out of the box with post mount thread issues, I think always on forks, but that's not something a consumer should usually have to deal with.
Post mount does not have IS's issue where the geometry is different front and rear. A +20mm post mount adapter does the same thing on both.
Facing a rear post mount on the chainstay, where they often are now on road/cross bikes, usually requires an offset facing tool to reach in and do it without bonking into the seatstay. If all manufacturers and dealers were well equipped and did their jobs right, this wouldn't make a difference for consumers. As it is I feel like there's a relatively high proportion of bikes like this sent out into the world still in need of rear post facing, but I don't have any proof.
Post mount has increasingly become the norm, probably because it's simpler, cheaper, and inherently more structurally efficient.
IS and post mount adapters are mostly dimensionally interchangeable between brands. It's wise when mixing brands though to take a close look that the pad contact looks how it should; weirdness does occur in this regard sometimes even though on paper it shouldn't be able to.
Flat mount is primarily for road bikes at present (some companies are experimenting with putting it on XC race bikes), more svelte and structurally efficient yet, and will probably end up winning out on road bikes. It was designed by Shimano and has now been adopted by everyone else. As the name indicates, flat mount puts the caliper closer to perfectly flush with the frame or fork.
Whereas IS and post mount take the same brake calipers for the most part (the exception being adapterless IS-specific brakes such as the aforementioned XTR BR-M975 and some Hope models), flat mount is a new caliper standard altogether. Adapters to put post-mount calipers on flat mount frames and forks exist, but in individual cases it's possible for hose/housing routing or frame clearance issues to make it ugly or unfeasible. Commercial adapters to put flat-mount calipers on post or IS mount and forks frames don't currently exist; spatial constraints may make such an adapter impossible or impractical.
Flat mount calipers have the mounting threads in the caliper itself. Flat mount only is compatible with 140mm and 160mm rotors for either front or rear. (It's likely that somebody will break this rule someday, as in the front particularly there's nothing physically preventing it.) There are two completely types of flat mount adapters, for front and rear. A front fork adapter (called a 'mount plate' in Shimano parlance) is always used. It's switchable for 140 and 160 by flipping it around. The adapter is first installed onto the caliper in the desired orientation using flush-mount bolts. The fork has threaded mounting holes that the adapter then bolts to through the holes at either end of the adapter.
In back, flat-mount compatible frames use bolts going through a frame member, typically the chainstay, to thread directly into the caliper if a 140mm rotor is used, or an adapter for 160mm.
An ironic point is that flat mount in an of itself is a design that did a good job of minimizing the number of adapters that may be needed, with the flippable front adapter that's always used and a choice of either no adapter or a +20mm adapter in back, for a very minimalist total of only two parts, but then there are also 4 different flat mount to post mount adapter caliper permutations (140 and 160 for both front and rear), which undercuts the minimalism if you're someone who has to worry about ordering the right thing or keeping all six in stock.
At the moment flat mount adapters are interchangeable and universal between brands as far as I know. Hopefully it remains that way, but there's nothing stopping someone from releasing a brake caliper that's an exception.