Cores and stringers and mesh...oh my!
Since starting eBodyboarding.com in 1999, I've fielded just about every question you can imagine and many you probably haven't about bodyboards. I am the one that answers all the Booger Picker submissions as well as recommends boards to customers in person here at our location and on the phone, so I'm pretty quick at determining what should work for a given rider. One thing that confuses a lot of new riders is the differences and effects of different core materials and stiffeners. Let's start by taking a brief trip down memory lane.
For the first 20 years after the bodyboard was invented (see our blog about this), polyethylene (PE) foam was the status quo as a core material. PE came in different densities and thicknesses. 
(A beautiful slab of PE core)
The core foam of the board generally determines the stiffness of the board, so as riders performed more demanding maneuvers, they needed stiffer and stiffer boards. The core foam got more and more dense, but that density reached critical mass. Any denser and weight and buoyancy would have been sacrificed. So, the magic density stayed with us through most of the 80s. The boards worked great, but the lifespan for an aggressive rider was limited. And since PE is a thermoplastic, meaning it's properties are greatly affected by temperature, warm water made them quite flexible, meaning it was easy to crease and buckle them in warm water and big surf. 
(Seriously creased board)
As a pro bodyboarder at the time, I was lucky to get a month out of a board during my winters on the North Shore. Luckily for me, I was allotted 24 boards a year if I needed them. So, how to solve the durability issue? There were plenty of experimental boards. Mike Stewart will tell you about some prototypes that he and Tom Morey worked on in the 70s that had stiff bottoms on them
(Tom Morey, always experimenting)
But ultimately, as I recall, it was the Morey Mach 10 that was released in the late 80s that had 3 flat graphite stringers inserted into the core that changed the game.
There was plenty of buzz about the Mach 10 and it's stringers, but at the end of the day it was still a PE core and in warm water, the flex was still there. The three "slats" of graphite made the board heavy as well. 
(The original Mach 10- Photo from VintageBodyboards.com)
Plus, those stringers really had a bit too much flex, much like a metal yardstick does. It was, again if memory serves, Buzz Morasca at Toobs who put out the first tubular stringer board. 
(Tubular stringer)
That paved the way for experimentation with stiffeners. Morey fired back with "X Flex Mesh" (which had the working prototype name "Performance Chicken Wire"). It was essentially exactly that...plastic chicken wire that was laid across the core on the deck or bottom side (or sometimes both) and then the skins were laminated to the core, sandwiching the mesh between the two. The effect was more "spring" to the board and prevention of over flexing. 
(Plastic mesh used in bodyboards)
This all helped, but ultimately in the quest for stiffer, companies like Challenger, BZ, Custom X and Morey started developing boards with Arcel, strand foam and ultimately, extruded polypropylene (PP). 
(The long-lost extruded polypropylene)
Technology with bodyboards hasn't gone much further since the widespread use of PP as a core material in the mid-90s and arguably took a small step backward when extruded PP became unavailable in the mid-2000s, and was replaced by a beaded version of PP which is now the core of choice for high-end boards. 
(Beaded polypropylene)
The flex properties weren't quite the same, so board manufacturers have taken to combining the PP cores with layers of mesh and 1-3 stringers to provide the perfect amount of flex. The end result is that boards today can achieve any level of flex you desire from the super-stiff to the nice and springy, and there are even interchangeable stringer boards from many manufacturers that allow you to adjust that flex by removing the stringer and replacing it with a stiffer or more flexible one. 
(ISS stringer system)
So, back to the purpose of this post. The long and short of cores is as follows:
These will have Expanded Polystyrene (EPS) cores. This is essentially styrofoam. It's neither durable or high-performance, but it's cheap. Most of these boards come with a leash too. If you're looking to just have a bash out there and aren't out to be the next world champ, this is your jam.
Typically as much as double the price of an entry level board or more, the PE core has stood the test of time for 4 decades as a great core that is both high-performance and buoyant as well as durable. Just keep in mind that it's a bit heavier than PP and will be on the flimsy side in warmer waters, so if you're an aggressive rider and you ride warm water zones (74 degrees and above), expect to buckle your board eventually.
(Flexing my board through my bottom turn back when PE cores were state-of-the-art)
Nearly all PE core boards, with a few exceptions, now have at least one stringer in them and some have mesh as well to prevent them from flexing too much, so even if you do happen to crease them, they won't hinge on the crease and will still work as well as they did before the crease. PE is the way to go for a high performance board at a reasonable price and in cold water environments is arguably the best core to have on your board.
PP BOARDS:
PP works well in all water temperatures. It's not as temperature-sensitive as PE, so if you want a board that will work year 'round or in the tropics, this is your core. It's a beaded core, so it "looks" like styrofoam, but it's chemical properties are a bit different, giving it more rigidity and a better spring. Every PP board on the market has at least one stringer and/or one to two layers of mesh for durability. The prices are higher for these boards, but you get what you pay for.
(A great demonstration of the difference in flex in different types of cores)
So, to summarize, which core is best is determined by your riding style and the water temperature you typically find yourself in, as well as your budget.
Something that's important to note, the foam for bodyboards is generally made by using a liquid plastic and blowing gas into it to create the air pockets to turn it into foam. Over time that gas will escape from the cells in the foam, rendering a board less responsive and less buoyant and give it a sort of waterlogged feel, so while you may have preserved that 20-year old bodyboard well on the outside, even if it's never touched the ocean, you can expect it to lack performance due to this aging process. For me, after about a year, I need a new board for this reason
For more information on board construction, check our Board Anatomy page!