For me, the best use is making beans. I find that home cooked beans have a much better taste and texture than from cans. The pressure cooker can make them in just about an hour from the moment you pour them in the pot to perfectly tender, and no presoaking is required, so I don't have to plan a day in advance.
Braises are, by their very natures, cooked to well done. They achieve an internal temperature well above 165 F which will make every meat well done.
Please see:
Why would a pressure cooker shorten a braise time?
for a detailed discussion on how the collagen to gelatin conversion takes place over time, and is made faster in pressure cookers.
The fact that the cuts that are suitable for braising are exactly the ones that allow the slow conversion of collagen to gelatin is what makes the pressure cooker suitable.
The real issue with cooking braises low and slow in the conventional oven is that at sea level, the internal temperature cannot get very high, so you need time for the gelatin to collagen conversion to take place. There isn't much point in raising the oven temperature higher than required to allow the time for this conversion, and it prevents overcooking or drying out the outside of the food while the conversion takes place. The meat will still be quite well done; it is only moist and succulent because of the melted fat, and the gelatin lubricating the meat fibers providing that slow cooked unctuousness.
In a pressure cooker, the ceiling temperature is raised, so the time can be shortened. The inside of the food is still quite well done. No loss to quality (since the food would be well done anyway), but a much shorter time.
If the food was not suitable to be cooked well done, the pressure cooker would be far from ideal, as it would certainly overcook the inside of the meat.
Note also that you have listed two special cases where foods are cooked at lower temperatures to avoid agitation (from the bubbling and boiling of the water), rather than because of issues directly related to time and temperature:
Stock. Stock is cooked at a simmer to avoid the turbulence and circulation from the bubbling and boiling leading to more dissolved, emulsified or suspended particles in the final product. That is, the goal is to keep the stock clear instead of cloudy. If you are not concerned with this aspect, it can be cooked at a full boil.
Beans are cooked slowly for several reasons, of which the main one is convenience. Cooking them at a full boil would require more attention (so they don't burn on the bottom), and is not terribly feasible in an oven, which is the easiest way to do them. Also, by cooking below the boil, there is less agitation in the pot, and so less splitting and sloughing of the bean skins, which some people find less than pleasing.
In the pressure cooker, you will not get this kind of agitation, because once the pressure is achieved and the food is at equilibrium, it is not going to be going at a full boil, but more of a simmer, but a much higher temperature simmer than is possible at sea level pressure.
Best Answer
Types
There are many different designs, but in general they all achieve the same result. The style you get comes down to personal preference
Personally I do not like the type with the lid that is captured under the internal rim on the pot, as they seem awkward to get the lid on and off
The quarter turn screw down lid (like a jar lid) seems easier to me
Pressure cookers work by increasing the internal atmospheric pressure so water has a higher boiling point, therefore you can fully cook something in water/steam in a shorter time frame. That's it, there is no magic. High pressure wet steam is very effective at transferring heat, so it cooks very evenly and rapidly. The pressure increase is not enough to significantly change the way things cook in terms of flavour absorption
Most pressure cookers come with at least two different pressure settings, medium pressure around 110°C, and high pressure around 120°C. Other than that, get the size you want
There are also the Electric pressure cookers, which are normal pressure cookers with built in computer controller heating and timing systems. These appear to be good idea, but the usually warning of fiddly electronics right next to heat and moisture apply. Some models have a variety of multi-stage cooking timers; including cooking while releasing steam (browning), keeping warm, time delay etc. A common complaint is that they cannot get hot enough, and are often rated at only 110°C
Cooking With
For cooking their main use is the "set and forget" type cooking, with the main goal being to save cooking time and energy consumption
Example #1: To cook a large (1Kg+) frozen corned beef, add about 500 ml water, two whole onions, spices and lock lid down. Place on stove on high heat until pressure come up (indicator pops out), and then set to medium heat and leave for a fixed time and it will be fully cooked with nothing raw or burnt. That takes about 50 minutes per 1Kg in our cooker; each cooker is different depending on exact pressure
Example #2: To cook rice risotto quicker. Use your pressure cooker pot to fry rice and get to the point you would normally start adding stock. Add stock and any other ingredients you would use in the first two thirds of your normal risotto cooking process time into the pressure cooker, lock the lid down, and quickly bring up to pressure. Cook for less than half the time that step would normally take. Run cold water over the lid to release the pressure and remove the lid, and continue your normal risotto recipe. This process can save a whole 10 minutes! Don't get too excited with this one, unless you have a lot of separate risotto batches to do, as you can bulk the pressure cooker step, and then finish a few separate risottos
Preserving With
In some cultures it is popular to preserve food using pressure bottling (canning). This should be the recommended method for anything without a high acid content (e.g. fruits)
Pressure cookers designed for preserving are significantly larger and stronger than their cooking counterparts, and will have a pressure and/or temperature dial on it. The idea being you can load a number of jars or cans into it at once, and bring it up to a higher pressure and therefore temperature. Most of these can be used for cooking too, if you have a large volume of food to do
For safe bottling (canning) of non-acidic foods you need to know that the entire contents of the jars have been brought up to a sufficient temperature for sufficient time to destroy all the bacteria. Typically this is just over 120°C for at least three continuous minutes for the entire contents of the jar. The whole pressure bottling process involves slowly and steadily raising the temperature to this point, and then slowly returning it to room temperature. Sudden changes in temperature will cause problems such as jar seal leaks.
Altitude has an effect on pressure bottling, and must be taken into account. Your local government food safety department should publish guidelines on the temperature and process suitable for your environment and culture
Notes
The main thing that tends to go wrong is that the lids pressure seal deteriorates over time (10+ years). So at the same time you but your cooker, order a spare and store in an airtight bag in a dark place until needed. They might not be available in ten years’ time!