The 10k glitch is a glitch that exploits how the game physics tries to slow Mario down when Mario is sliding forward and the player tilts the joystick backwards. For example, on a default friction floor type, Mario's speed is multiplied by .92 + (.01+v/10,000)cos(DYaw), where v is the speed units Mario has. DYaw is the difference in angle between where Mario is facing and joystick tilt, thus cos(dyaw) is always negative. The slowing effect is stronger if Mario is faster. If Mario has 9100 speed, he will immediately stop in place. But past that, Mario's speed can be instantly reflected into small backward speed, and once he has more than 20k speed, Mario's speed can even multiply by several factors. At 90k speed, Mario's speed is scaled by -8 into -720k speed. And at 262k speed, his speed is scaled by -261 into -69.4 million speed.
To perform the 10k glitch, Mario must be in a braking state (holding backwards on the analog stick as if he were about to sideflip) with high forward speed (at least 20k) and then touch a slippery slope. Since most courses are too small to attain this much speed while touching such a slope, this glitch is typically performed using PU speed.