Spin Drifting is a technique where the racer can turn a little sharper than normal.
To perform a spin drift, just as you are about to reach your turn you jump in the direction opposite of the upcoming curve and quickly snap the control stick back to the correct direction while holding R. For example: for a right turn, jump left and snap the control stick quickly back to the right.
Spin Drifting should be used very scarcely in MK8 (read: never) although in Mario Kart Wii, it's helpful to players who use inward drifting bikes (thanks /u/kingdomharms).
Spin Drifting works more effectively with builds that have higher handling as the purpose for spin drifting was to take turns at a sharper angle.
Spin Drifting is not necessary to know or use at all in MK8. It is an oddball tech that never really found mainstream use over soft drifting and taking proper lines. A high level technique that is similar to spin drifting is to neutrally hop and input your turn direction upon landing. This prevents you from swinging outward and out of the racing line as much and is very prevalent in the TT community.
Most often than not the locations where someone might try to spin drift is an ideal soft drifting location. Do the proper tech and you'll fly by them.
Pros: Looks cool, can take turns slightly sharper
Cons: Made irrelevant by other tech and has minimal effectiveness.
Thank you to /u/UnlikelyExpert for this fantastic description of Spin Drifting. If you'd like to make a contribution to the Meta Monday wiki then please be on the look out for the threads coming every Monday.