A spring reverb unit is a piece of equipment that attempts to create an artificial reverb in the following way: Sound is sent through a transducer to a spring (the transducer in this case is something that converts the electrical energy into mechanical energy, similar to the driver of a speaker). The spring vibrates with the sound. The nice thing is that the spring does not stop vibrating with the sound, but retains some residual vibrations. Another transducer at the other end of the spring (which in this case works like a pickup) converts the spring vibrations into electrical energy. This transducer picks up not only the original vibrations created by the first transducer, but also the residual vibrations of the spring. These residual vibrations are what makes the signal sound reverberated.
Spring reverb is reverb simulated by using an output transducer to pick up the residual vibrations of a spring after initial vibrations in the plate are generated by an input transducer.
If you used a metal plate instead of a spring you would get a plate reverb unit. Even though both are constructed similarly, they have different qualities. The spring reverb has less diffusion and absorption and different coloration – it accentuates mid to low range frequencies (under 1 KHz).
In the digital world actual springs do not exist, but digital reverbs can attempt to model these characteristics of actual spring reverb units and so you may see "digital spring reverbs".