Hi Ginger,
There are many different styles of mattress and each design can have advantages and disadvantages. Everything involves a tradeoff of some type. For example a two sided mattress has a durability advantage but it also limits the design and layering of the mattress to some degree and is more expensive than an equivalent one sided design.
A mattress with box springs (usually but not always this is an innerspring mattress or a thinner foam mattress where the box springs add to the designed performance of the mattress) can be two sided or single sided and an all foam mattress that uses a foundation instead of box springs can also be single sided or double sided.
They key word in your question is “quality”.
No matter what style of mattress you choose the important part is that it meets your pressure relief and alignment needs and all the preferences that are important to you (there is more about this in post #2 here) and that it uses the highest quality and most durable materials that are available in your budget range.
Phoenix