It's pretty much the most basic thing about going fast - how much power you can keep on the pedal for the most consistent amount of time.
DOWN - For the most part, and where the most power comes from, you simply press down on the pedal.
UP - When you need more power, or while sprinting or climbing fast, pull up on the pedals when you're not pressing down. You can get a lot more power this way.
ANKLING - This is when you just need that little extra, usually to avoid getting dropped. Basically you just press the front of your foot down at the end of the stroke, kind of like pressing on the gas pedal of a car. The power originates from your calves.
PULLING - This is best when the pace is consistently really high. This replaces ankling and is between the down and up stroke, and it's a movement similar to cleaning the bottom of your shoes on a door mat.
The magic number for how fast to pedal is 90rpm. Too much slower than that you're mashing on the pedals and will fatigue your muscles quickly; too much faster than that and your butt will be bouncing around in the saddle.
Some people look super smooth, other people look like they're struggling with the bike. It all comes down to having a solid and consistent pedal stroke at a consistent cadence.
Standing is a whole different story and deserves it's own post.