There are a number of techniques and materials that you can use to make inflatables ranging from the relatively inexpensive to quite costly.
There are two main types of inflatables, cold air and sealed. Most inflatables that you see are cold air inflatables. Sealed are inflatables are inflatable boats, balloons, air mattresses.
Cold Air Inflatables
Cold air inflatables require constant airflow but for a smallish inflatable a very small and relatively quiet fan can be used. Also, if the form will be round it will not need a lot of static pressure to keep its form. Most lightweight cold air inflatables are sewn together out of nylon or polyester fabric ranging from 70 - 200 denier. The fabric has a coating on one side, often a urethane coating. This coating helps keep the air from passing through the fabric. Bonded polyester thread of Tex 69 - Tex 92 size is commonly used with double stitched seams to sew inflatables. This type of cold air sewn inflatable is durable and long lasting, especially indoors. Clear vinyl or translucent fabrics can be used along with interior LED illumination. If you use clear vinyl it is important to use a long stich length. As you stich on the vinyl you are basically perforating the fabric and making it weaker. As you increase the stitch length, use a thicker thread.
Sealed Air Inflatables
Sealed air inflatables do not require constant inflation but will always need to be topped off every few days / weeks depending on the quality of the seals and construction. Think of a pool toy or air mattress. These are usually made with specialty equipment including thermal and sonic welders.
DIY Sealed Air Inflatables
You can make DIY sealed air inflatable with common household equipment. Polyethylene sheeting from the hardware store can be thermally connected to itself. A household iron works for applying enough heat to seal the plastic.
If you place the iron directly on the fabric then the plastic will melt to the iron. Parchment paper can work as a barrier. I usually fold a piece of parchment paper in half. Then I trim it to the width of the iron. I place the two piece of plastic sheathing in between the layers of parchment paper. Then I run the iron on top. It takes a bit of trial an error to get the correct heat settings and speed for the plastic you are using to make your inflatable.
1 mil polyethylene sheeting is too thin for most projects. 3 mil is a good balance between weight, durability and cost. 6 mil is the most durable but can take more heat and time to seal completely.
Since an iron is not a perfect thermoplastic welder, make sure your seams are thicker than needed to account for weld imperfections.
If you need to seal on top of an already closed form such as a tube, you can put a piece of cardboard and parchment paper inside the tube. This will prevent the top layer of plastic from sticking to the bottom layer.
Getting Started
I recommend that you get some plastic, tape and a fan. First make a simple “pillow case” form. Then attach the fan and blow it up. The next task would be to make a cube or retainer prism form and attach a tube to it. Making a cube is much more complex since it takes 6 pieces and you need to think about how to finish the seams on the corners. Attaching a tube is straight forward if you remember C = 2πr to calculate the circumference of a circle.
Use this after you make the tube. Make the tube flat and measure the C of the tube to calculate the r of the tube. Then you have to subtract your seam allowance from r. Then you know the r of the hole you need to make. The hole will be smaller due to the seam allowance.
After that it is variations on a theme only limited by your imagination.
Remember that to have a concave shape you need something inside the inflatable to hold the sides in since all inflatables want to be round and convex.
Price vs Quality
On the low end you can use 2-4 mil polyethylene plastic sheeting and heat seal it together with standard clothing irons. If you put parchment paper above and below the plastic then it won’t stick to the ground or the iron. You can also use packing tape and / or duct tape to seal the plastic together This can make a very large, strong, lightweight inflatable cheaply. You need a fan to blow air into it constantly. For the fan you can use a simple box fans or HVAC duct booster fans. It is best to fabricate a tube of plastic coming from the inflatable that is just a bit bigger than the fan so you can easily install the fan.
Next step up in quality and price is to use nylon or polyester fabric and sew it into the shape that you want. This is more costly and a bit more technical but is great if you want to reuse the inflatable more than 5-10 times. It still needs a fan to blow constantly to keep it inflated.
Heavy Duty
Bouncy houses are made of PVC coated polyethylene mesh. This can be sewn or heat welded together. It is very strong and durable but expensive and heavy. It requires some specialized equipment to heat seal it together.
PVC coated polyester can be glued with HH-66 vinyl cement or hot air welded together. This requires proper ventilation and safety equipment.