These batteries can be separate AA (or other size) batteries that you charge in a separate charger, but they also include the battery in your cellphone, laptop computer, media , portable game player, cordless electric shaver, cordless drill, etc.
This ad is inaccurate. Cardboard, glass and plastic CAN be recycled. Rechargeable batteries NEED TO BE recycled. They contain toxic materials. These components should neither be burned nor added to landfills. In addition, they contain rare materials, like lithium and cadmium, that need to be reclaimed and reused.
Even if you don't recycle in general, rechargeable batteries need to be recycled. Most stores that sell them have drop-off locations.