Long before the invention of the ollie, a skater named Curt Lindgren invented the original kickflip. It was probably in the mid-1970s.
Freestylers still like this move and it's many variations. There is the 180 kickflip, the helicopter, M-80 kickflip, etc. Learn this one first, and you'll have lots of fun learning the rest!
Riding forward at whatever speed you find comfortable, move so that your feet are parallel, in the middle of the board. Then use whichever foot feels right and hook your foot under the board. Keep the heel on top of the board, and just move the inside edge of the shoe down. Your weight on that foot is on the heel, which is still on top of the board.
As you do this, compress for a jump.
Now jump. As you go up, use the foot that is hooked under the board to give the board a flick, sending it into a flip. Don't put the flipping foot on the ground. By having that heel on top of the board you can get plenty of flipping power. Keep your center of gravity over the board.
In frames 3 and 4, I'm still moving up and the board is flipping. I'm watching the board flip, waiting for it to come back around for a good landing.
In frame 5, I begin to put my feet back down to catch the board as the top comes around. In frame 6 I land, with my knees bent.
Try to land with your knees bent a little. If you straigh-leg the landing, you'll probably land on your butt.
Now just roll away. Practice about 1000 times, and you'll have it.