Plastics are called plastic because they are pliable, namely, they can be shaped and molded easily. As plastics become easier to mold and shape when they're hot, and melt when they get hot enough, they are called thermoplastics. This name can distinguish them from crosslinked materials that don't melt, which are called thermosets.
Well, why a material is called a plastic and not a rubber or elastomer? The answer is in the bouncing. An elastomer can be stretched, and bounce back. Plastics tend to either deform permanently, or just plain break, when it is stretched too hard.
Although plastics don't behave as well as rubber when they're stretched, it takes a lot more energy to stretch them in the first place. The fancy way to say that is "plastics resist deformation better than elastomers do". This is good when the material doesn’t need to be stretched.
In fact, it takes more energy to stretch the plastic, making it resistant to deformation. But at the same time, if we pull hard enough, we can not only stretch a plastic, but it will stay in the shape we stretched it into once we stop stretching it. Elastomers bounce back when we let go. Plastics are also much more pliable than some other materials, like fibers. Fibers stretch very little when we pull on them. This makes them good for things like rope.
We can find both hard and soft plastics around us. The plastic keys on the keyboard are hard, while the plastic around the cables of the same computer is soft. This is because all plastics have a certain temperature above which they are soft and pliable, and below which they are hard and brittle. This is called the glass transition temperature, or Tg. The Tg is different for each plastic. At room temperature, some plastics are below their Tg, and so they are hard. Other plastics are above their Tg at room temperature, and these plastics are soft. Sometimes additives are added to a plastic to make it softer and more pliable. These additives are called plasticizers.
Some polymers used as plastics are polyethylene, polypropylene, polystyrene, polyesters, polycarbonate, PVC, nylon, and poly(methyl methacrylate).