The most important thing to remember is to prevent this situation. You should never find yourself in the position to slip down a slope. In many cases (steepness, snow/ice conditions, ...) there is no way that you will stop once you are slipping, even if you execute the following perfectly. Still we want to be prepared for the worst case as well. And knowing the following is just a first and minor step, the most important thing is to actually try it over and over again. Just make sure the terrain is suitable, so no rocks and a flat (snow) section at the bottom.
There are two phases: First you need to get yourself in a stable position, then you need to brake until you eventually stop. The techniques vary depending on the equipment: With an ice-axe and crampons, with an ice-axe only or with neither.
The final position you want to be in is front to the ground and head up-slope (against the direction of slipping). Details vary depending on equipment.
Ice-axe and crampons
Basically the same as with just an ice-axe, however you must make sure your feet are off the ground at all times. Your crampons catching potentially results in uncontrolled tumbling and serious injury.
Turn sideways, if on your back.
Plunge the pick of your ice-axe into the snow. Do it gradually and keep a firm grip. If you do lose the axe, proceed as described below. This will automatically turn you in the head up position.
Essentially the same again, but instead of plunging the pick into the snow you brake by pushing your arms into the snow. Just try to lift your upper body with the arms, as if you were going into a plank position or doing push-ups.
Ice-axe and crampons
Still keep your feet away from the ground. If the terrain permits it, keep that way till you stop; it's the safest. If the terrain becomes rocky, try to brake without your feet until there is no other option any more. Meaning the risk to get into an uncontrolled tumble is negligible, as falling over a ridge/crashing into rocks/... is imminent and worse.
Hold one hand on the shaft, the other on the head of the ice-axe and position it in front of your chest. Apply pressure by leaning onto it. Again do this gradually, sticking it into hard packed snow will probably yank it out of your hands.
This position might be scary as the ice-axe is pretty close to your face. True, this is dangerous, but this is a life-threatening situation, so braking efficiency wins over the potentially broken nose or missing tooth.
Get up onto your underarms, then lift all your body so that only the underarms and feet are in contact with the snow. In certain conditions you might even want to get into a push-up position (only hands and feet on the ground).