Hi Eloise;
Julian made a good start, I'll try to expand it for you.
We will start with J-Tubes the "original" RMH.
Wood is fed vertically into an open feed tube.
With an 8" build you can expect to add wood every 45-60 minutes.
A J-Tube was traditionally plumbed through pipes embedded in a solid mass of stone and
cob usually in a long bench.
With the advent of Bells (stratification chamber) a J-Tube can be plumbed into a hollow bell and made more compact than a long bench.
A disadvantage to a J-Tube is the open feed tube. Children will drop toys and other things into the fire... some will attempt to reach in and remove the burning barbie doll...
Cats absolutely love any RMH, but with the open feed tube, long bushy tails could become warmer than intended, and your home could be in fire danger if things deteriorate.
Walker stoves) Matt's stoves feature his "riser-less" cores.
His best-known stoves are cook stoves, although he does offer J-Tubes and bell-designed BBQ ovens.
His stoves are suggested to be built with ceramic fiberboard, although he also allows firebricks to be used.
Ceramic boards are insulating, they allow the heat created to pass quickly through the stove making for an easy almost smoke-free start.
The problem with ceramic boards is how very fragile they become after heating. Adding wood to a fiberboard firebox it is very easy to damage and requires it to be replaced.
Matt recommends lining his ceramic firebox with expensive RA330 sheet metal to protect it from abrasion.
If you choose to build Matt's stoves with firebricks, then you have no worries about abrasion, you will however struggle to build a fire without smoking out the room.
Until your firebricks heat up, the brick riserless core has trouble drafting, a bypass is required to assist in getting things going.
The firebox on a Walker stove is small, so it requires more tending than a Batchbox.
Batchboxes) although originally developed by Lasse Holmes from Homer, Alaska.
Peter Berg, has further tested, refined, and developed several design variants of the Batchbox.
The first-generation batchbox is the most well-known and the easiest variation to build.
Batchboxes load wood traditionally, they have metal doors.
Dragontech-designed doors for first-generation batchboxes are spark-free and have a small viewing window.
Batchboxes have a large firebox and can be expected to burn and have coals for two-three hours.
Batchboxes are also the hottest burning with riser temperatures over 2000F.
Batchboxes are almost exclusively plumbed into bells rather than a piped mass.
Batchbox variants)
First generation, easiest to build
The DSR variations 1,2 and 3 are tricky to build but have an upper viewing window with beautiful double ram horn flames to enjoy
The Shoebox design is similar but has no window (I think)
Peter's newest design core is the Shorty core, which is not difficult to build, but requires three large slabs to be poured at home.
I built and installed the first Shorty core in the USA.
It utilizes a unique airframe that supplies fresh air to the top and upper sides of the batchbox rather than the air coming in from the bottom like all other wood burners.
This airframe allows the use of a door with a large window, (mine is 7x7)
Note) In 8 weeks of use the window glass has never gotten covered with soot, that is how clean-burning Shorty is!
In my opinion, Shorty is Peter's crowning glory! (I also have two first-generation Batchboxes and a Walker riserless BBQ)
Almost impossible to overfuel, Shorty appears to just be casually burning.
Looks can be deceiving! Shorty is every bit as badass a
dragon as any of the earlier variants, she just does not feel the need to showboat and fly around your valley roaring!