What is the name of this springy type oscilloscope probe accessory?
It's a "Probe tip ground clip" (or something very similar like the examples below)
An example of it being named as such is here. It's called "Probe tip ground" here. It is also called a "Probe tip spring adapter" here.
Whatever it's exact name, I think the main thing is to have "tip" in there, as it differentiates from the standard ground clip by implying the clip is used at the probe tip.

Basically, it's function is to shorten the ground lead, as this lowers inductance and enables higher frequency signals to be displayed accurately.
Try it with a very fast rising signal, and compare it to using the normal ground clip and you will see the difference. A common mistake is using the wrong setup to probe a signal and then assume the fault is with the signal, when it's really issues caused by the setup. 

The way to use it is like the following picture, touch the ground wire to the IC ground pin, and the probe tip touches the pin you wish to probe:

 

When designing your board, you can add dedicated test points to make getting to the signals easier. You can even add on board probe tip adapters (available from any decent distributor) if you have enough space.

For a detailed discussion of scope probes the Tektronix ABCs of Probes is good reading.
Here is a diagram from the above link showing the effects of different ground lead lengths on a fast rising signal: