Polyurethane (also called polyfoam or just poly) is the most common type of foam you will see in mattress comfort layers and it is used in almost all mattresses made by the major manufacturers or sold in larger outlets. It is common to see it used as a very top layer over other higher quality foams used in comfort layers. The reason for this is that it is very inexpensive to manufacture in its lower grades (which are almost always used outside of mattresses made by higher quality local and regional manufacturers) and it can be made to be very soft which can make a mattress leave a very comfortable initial impression in the store. These together equal higher profits ... at the expense of long term suitability, durability and satisfaction.
The widespread use of lower quality soft polyfoam in mattress comfort layers that are too thick is perhaps the single biggest source of consumer complaints and dissatisfaction today. It will almost always be the weakest link in your mattress and quickly lose the qualities that made it feel good initially and wear out more quickly than other parts of your mattress. It is also the source of the widespread complaints about body impressions that are common to ALL major manufacturers because of the use of this material (or synthetic fibers) in the comfort layers of a mattress. You will often read about or be told that body impressions of less than 1.5" - 2" are completely normal and while this may be true with natural fibers that rely to some degree on body impression to provide pressure relief, in a foam it is a sign that the foam is wearing out and is not "normal" at all no matter how common it may be. Long before you see deep depressions in a mattress (which is a sign of complete failure in a foam) the foam will have lost most of the qualities that it needs to provide pressure relief and comfort. Worse yet, mattresses that are sold as being "latex" or other higher quality materials will usually have several inches of soft lower grade polyfoam or synthetic fiber on top and when it wears out or forms impressions, the consumer often does not realize that it is the polyfoam that is wearing out rather than the higher quality foam beneath it as they didn't even realize it was in their mattress.They may then come to believe that the higher quality material in their mattress wore out too quickly and replace it with a polyfoam mattress ... and the cycle repeats.
Polyfoam is chemically similar to memory foam except it doesn't have the added chemicals that make it soften or "melt" under pressure or heat. Because of this it will compress and rebound instantly instead of "flow" under pressure and is more resilient than than memory foam. It comes in 3 basic grades and beside the issues of durability, the lower grades don't spring back nearly as well (they are less resilient) than the higher grades. All the grades can be made in "softer" or "firmer" versions (although the middle grade HD tends to be more commonly available in firmer grades) so "softness" or "firmness" is not an indicator of quality. The grade of polyfoam is measured by its density or weight per cubic foot and by its "support factor" or progressive resistance to compression.
Regular or conventional polyfoam: This is the lowest grade of polyfoam and weighs less than 1.5 lbs per cubic foot. It is the least expensive grade of polyfoam and is not generally suitable for use in the comfort layers of a mid to higher priced mattress that has any expectation of lasting a reasonable length of time under regular use (outside of an inch or so in the quilting). It is an inexpensive (cheaper) material that can be made to feel very soft though and feels good in thicker layers when you lie on a mattress in a store. Because even ultra firm mattresses need something soft on top of the support layers ... and this is often the least expensive way to do it ... you will mostly see this lower grade soft polyfoam to some degree in almost all "national brand" mattresses made by major companies or sold in larger stores. It is quite simply not suitable for use in a mattress that is used regularly and carries a mid to high price tag no matter what "story" about other materials may be attached to the mattress itself. When mattresses were more commonly 2 sided, at least each side had a chance to rest and didn't break down quite as quickly but with one sided mattresses combined with low or mid grade soft polyfoam in the comfort layers (or worse yet in pillowtops that are thicker) ... you can quickly have depressions in a mattress that uses this material. The lowest grade of conventional polyfoam (1.2 lbs and less) is not even suitable IMO for use in a "throwaway" mattress at the lowest budget level. The only exception I would make would be a thin quilting layer (in the range of around an inch or so or less) where foam softening will have much less effect and can provide a surface feel over higher quality foams that many people like.
High Density polyfoam (HD): This is polyfoam that has a density of 1.5 lbs per cubic foot or more although 1.8 lbs is a better guideline and would generally considered to be the bottom end of high quality polyfoam. It is sometimes difficult to find this in higher densities in a softer version but it is quite commonly used in the comfort layers of a mattress. It will last longer than the lower density conventional polyfoam, has better compression and performance qualities, it is suitable for use in a comfort layer when used reasonably (layers that are not too thick) or with slightly firmer foam. The lower end of the scale especially (1.5 lbs) should be used in thinner layers or in a two sided mattress to improve durability and reduce the effect of foam softening while in a one sided construction I would use 1.8 lb density (or preferably 2 lbs with higher weights) as a minimum. When purchased from a local manufacturer who is transparent about the density of the polyfoam and construction methods they use, this grade of polyfoam in slightly firmer versions can be a good comfort layer choice in relatively low to mid priced mattresses and still provide good durability. In very thick and soft versions or when the construction methods do not support durability or are unknown it should be used very sparingly or at least there should not be an expectation of good durability attached to it.
High Resiliency polyfoam (HR): This is the highest grade of polyfoam and weighs 2.5 lbs per cubic foot or more. To qualify as HR it must also have a support factor of 2.4 or higher. This support factor is an important part of why a HR polyfoam has the qualities it does as it is made with a different chemical formula and has a different cell structure than the two lower grades. I would also encourage you to read the section on polyfoam as a support core section as HR polyfoam is one of the most mislabeled materials in a mattress ... both in stores and on the internet. Because of its different formulation, it is more durable in all ILD's from very soft to very firm and is suitable for use in any layer of a mattress including comfort layers. It is more expensive than the lower grades though so you rarely see soft HR foam in the comfort layers of a mattress except in those made by smaller local and regional manufacturers as it does not have the same profit margin as the lower grades that are sold with a "story" attached. Some of the best HR foams come close to latex and are good quality, have good resilience, and will last for many years.
Unless you specifically know the quality/density of polyfoam in a mattress comfort layer or quilting, ... it is usually wise to assume that it is a low grade and anything more than about 1" or so should be avoided in either the comfort layers or the quilting above it. Once you are at about 2" or more you have a construction that will givie you a "head start" in the formation of impressions and can reduce the performance of your mattress as it softens and wears out. Unless you are certain that any polyfoam in your mattress comfort layer is HR (or HD with the appropriate construction methods and grade for the price) and have been given the exact specifications of the foam from a source you trust (such as a local manufacturer), minimizing the use of polyfoam in the comfort layers is usually your best choice.