There are many thousands of different mattresses that are sold in the industry that use a wide range of different components and materials in their design which means that it can be confusing or even overwhelming to try to figure out what two different mattresses have in common but in very general terms they can be categorized into a a much smaller number of categories based on the primary materials and components that are used in the support core and the comfort layers of the design. While each category can also include hundreds or sometimes thousands of individual models with a very wide range of thicknesses, layering combinations, and firmness levels ... knowing the most common categories of mattresses can sometimes be helpful if your testing indicates a "pattern" that you tend to prefer some types of mattresses over others or in making more "apples to apples" comparisons between mattresses. The "definition" of a category is generally based on the type of support system and the type of comfort layers that are used inside a mattress.
I would also keep in mind that the choice between different types of materials and different types of mattresses is primarily a preference or a budget choice rather than a "better/worse" choice and different people will prefer some types of materials or mattresses over others. Each type of material will tend to have lower quality and less durable versions and higher quality and more durable versions so no matter which types of materials or mattresses you tend to prefer I would always make sure you know the specifics of all the layers and components inside any mattress you are considering (particularly in the upper layers of the mattress) so that you can confirm that there are no lower quality and less durable versions of any particular material in a mattress that could be a "weak link" in terms of it's durability and useful life and how quickly you may need to replace it (see this article)
These are the dozen or so most common types of mattresses that you will find in the industry.
Standard innerspring mattress: These are still by far the most common mattresses sold in the industry and are still very popular. They all have some type of innerspring as a support system and polyfoam or natural or synthetic fibers in the comfort layers. They have the familiar "feel" and "bounce" of an innerspring which can also be helpful for the "other" activities that happen on a mattress. They can range from very low quality "cheap" mattresses that use lower cost innersprings and lower quality and less durable materials in the padding to some very high quality and more costly mattresses including some that are two sided. The type of innerspring and the layering above the springs will make a significant in the feel performance of the mattress and in the amount of motion transfer of the mattress as well. The weakest link of most traditional innerspring mattress will tend to be the quality/density of any polyfoam above the spring ... not the innerspring itself.
Innerspring/natural fiber mattress: These types of mattresses used to be much more common before polyfoam became the dominant padding material in mattresses. They range from relatively lower cost innersprings with cotton padding (often two sided) that are usually on the firmer side to some ultra premium hand built mattresses that use pocket coils in combination with multiple layers of more premium fibers such as wool, horsehair, and cashmere that can sell for many thousands of dollars. Because of the natural fibers in the comfort layers and the innerspring support cores they are among the most breathable and temperature regulating mattresses in the industry.
Memory foam mattress: These mattresses typically use anywhere from a single 2" layer of memory foam (or gel memory foam) to multiple layers of different types and thicknesses of memory foam on top of a polyfoam support core. They can also range from very low quality and "cheap" mattresses with very little durability to some very high quality mattresses as well. If you are considering a memory foam mattress then in most cases the most important factor in its durability will generally be the quality/density of the memory foam layers about the polyfoam base layer.
Innerspring/memory foam hybrid: This is a "hybrid" mattress and would be attractive to those who like the feel of a memory foam sleeping surface but also prefer the feel and more lively response of an innerspring support core. While the main material in the comfort layers will be memory foam ... they will often include other materials in the mix and I would be very cautious to make sure that the memory foam itself is a high enough density to be a durable choice and that the other materials in the mattress are also high enough quality/density that they don't become a weak link in the mattress.
Innerspring/latex hybrid: This is another "hybrid" construction and is very popular among some of the most knowledgeable people I know ... particularly with a pocket coil support core and latex comfort and transition layers. They have most of the benefits and durability of sleeping on latex and have the more familiar feel of an innerspring as well. They can be a very good choice.
Polyfoam/latex hybrid: This is also a hybrid mattress that usually uses at least 2" of latex in one or more layers on top of a high density polyfoam support core. It would be attractive to those who like the feel of sleeping on latex comfort layers but aren't in a high enough budget range to be able to afford an all latex or mostly latex mattress that uses a latex support core as well. It will have some of the "feel" and benefits of an all latex mattress. I would make sure that the base layer is at least 1.5 lb density if you are looking at lower budget ranges, 1.8 lb density if you are in more average or higher budget ranges (which would be more durable) and 2 lb density if you are in a higher than average weight range (which would be more durable yet).
All or "mostly" latex mattress: These are generally very high quality mattresses and can be among the most durable mattresses in the industry. Latex is the most resilient and "springy" foam material and other than innersprings are the most lively and responsive type of mattress so they can also be great for some of the other activities that happen on a mattress as well. There is a wide range of latex designs that can use either Dunlop or Talalay latex using natural or synthetic latex or a blend of each. While latex is a very high quality and durable material ... it is also a more costly material than other types of foam as well.
Coil on Coil mattress: A coil on coil mattress will typically use an innerspring support core and then a thinner version of a pocket coil called a microcoil as a comfort layer. The innerspring support core can be any type of innerspring (Bonnell, Offset coil, Continuous coil, or a Pocket Coil) and there are various types of microcoils that can be used on top with different gauges, heights, and coil counts. Microcoils are also a durable component and provides a more responsive and lively feel than most foam materials that many people like. These mattresses will generally uses other types of foam either below or above the microcoil as well to even out the response of the microcoil springs so it's important to make sure that these other materials are also good quality and durable materials and aren't a weak link in the mattress.
All polyfoam mattress: Polyfoam in general is one of the lowest cost materials in the industry (although there are a wide range of different grades and prices for polyfoam) so these are usually lower budget mattresses that will include one or more layers of polyfoam in their design. They will often have a support core of high density polyfoam and then one or more comfort layers of softer polyfoam as well that can range from high density, high resilience, or high performance polyfoam. With this type of mattresses it's especially important to make sure that the quality/density of the foam is durable enough for your body type and for the expected useful life of the mattress because they can range from low quality/density and less durable versions of polyfoam to higher quality and more durable versions that will last for many years. There are some really "cheap" mattresses in this category that I would avoid completely and there are also some higher quality mattresses in this category that would make a durable choice in a reasonable price range.
Airbed: See this article.
Waterbed: See this article.
Other mattresses: This is a catch all category for some of the less common mattresses in the industry that don't easily fit into another category either because they don't use the more common materials and components or because they use a mix of materials in the comfort layers without any of them being dominant or more than about 2" thick Some examples would be a wool mattress, a mattress that uses buckling column gel in the comfort layers, or a mattress that uses a combination of memory foam and latex layers that are each less than 2" on top of a polyfoam or innerspring support core.
While each of these categories have a very wide range of different models and designs and can be anywhere from very soft to very firm ... by knowing the different mattresses categories it may help you to identify any general patterns in the types of mattresses and materials that your testing indicates you tend to prefer and help you narrow down your choices a little more easily.