There are a lot of different styles of cooler. But they all have some common features. First off, there must have a layer of insulating material to keep the drinks cold. Then on both sides of the insulating material you need a layer of waterproof material to keep it sealed. Then you need a method of attaching all the layers together. Lastly, you need a way of opening and closing the cooler.
Waterproof Material: Generally sheets of thin plastic work well. If you want to make a flexible cooler, then I recommend using a material like vinyl. This can be purchased by the yard at fabric stores. You can also use cheap plastic shower curtains or plastic table cloths.
Insulating Material: Any insulating material can work. Flexible foam is probably the easiest to work with. You can probably find a suitable material lying around your house. You can use packing foam, craft foam, or even bubble wrap.
Attaching Material: You can either attach the waterproof material directly to the insulation or you can attach the two outside layers together and create a pocket with the insulation inside. You also have a number of options in how you attach the layers. You can sew the layers together. This is probably the best option. It gives you strong uniform edges. However if you aren’t skilled at sewing, you have other options. You can use glue or tape (be sure to use a glue that is designed for the materials that you are using). If you are using plastic, it is also possible to melt the layers together. However this can be quite difficult. It takes a lot of practice to be able to properly melt plastic together without messing it up.You have a lot of options. Feel free to do whatever is easiest for the materials that you are working with.
Opening/Closing Mechanism: The most common methods for closing a flexible cooler are either zippers or Velcro.
You have a lot of options in how you construct your cooler and as a result you have a lot of options when choosing what materials to use. In this example, I am making a flexible cooler with craft foam, a vinyl table cloth, spray glue and Velcro.