A wine corker is a specialized tool that allows you to insert a cork into your bottle in a quick, easy, and safe manor. Corkers work by first compressing the cork, then inserting it into the bottle in a single motion. All you need to do is place a cork in the mechanism, position the bottle correctly and pull the lever(s) down until the cork is fully inserted. Voila, the bottle is sealed!
Once sealed, the bottles should remain upright for 2-3 days to allow the pressure in the bottle to equalize and the cork to fully conform to the neck of the bottle. If you are using natural corks, then after these 2-3 days go ahead and turn the bottles on their side and store them that way until the wine is consumed. (Storing the wine on its' side helps keep the corks moist by being in contact with the wine. This ensures that the cork will not dry out, shrink or crack which would allow oxygen in and ruin the wine). If you are using synthetic closures, however, the wine can be safely stored upright and does not require being stored on its side.
A quick note on using synthetic closures:
Hand corkers are not designed to work with synthetic corks. If you wish to use synthetic closures then you will need to use a floor corker to do the job.