According to HowStuffWorks, the substance that gives bubblegum its chewy nature is a natural or artificial form of rubber. Originally, most bubblegums used chicle, the sap of a cousin to the rubber tree, but many modern gum varieties use a synthetic gum base.
The rubbery gum base inside bubblegum is what gives each piece its soft and chewable texture, but also prevents it from breaking down completely when chewed. In addition to being tough and resilient, it is extremely stretchy, allowing you to flatten it out and inflate it into a bubble with ease. The flavor of "standard" pink bubblegum is actually the taste of the synthetic gum base itself, since the materials used naturally produce a sweet flavor.