Your first motion is to take the stone and pull it backwards, frequently lifting it off the ice in the backswing, then you swing it forward into a smooth glide down the ice. However, to deliver a stone well, you should glide along with the stone as far as possible. Good curlers usually glide very close to the ice, in an odd, very stretched pose. As you slow down, the stone glides on, to come into play on the opposite side, beyond the other hog line. The Game Curling is always mentioned as a game of strategy by curlers, partly because it is, but also probably because they want to make sure that it's seen as more than throwing rocks and slipping around on the ice.
Strategy is definitely the big thing in competitive curling, though.
A great deal of effort goes into planning an end so your team's stone ends up closest to the center. You send the stone down the sheet, and with the help of the sweepers and the direction of the skip, you somehow get the stone to stop where you want it. Here is shown a perfect draw into the Tee, the center of the House. This would be fairly pointless as a first shot, as it could easily be taken out. Here, the green stone is taken out by the yellow.