First you must buy a ship that will support a Fighter Hanger. The updated list is here:http://elite-dangerous.wikia.com/wiki/Fighter_Hangar
Second you need to buy a fighter hanger and a fighter for the hanger. Do this in the Outfitting screen.
Third, you'll want to hire a crew member and add them to your ship. Do this in the Crew Lounge screen.
Now when you are out flying around in normal space (not Supercruise or Hyperspace) you can use X then down on the D-pad to select your fighter and launch it. You will have the option of assigning your crew member to the fighter or leaving them to fly your main ship while you personally pilot the fighter; either will work.
[There used to be a tricky bit (bug?): You had to pilot the fighter yourself for this achievement to unlock. Sending out your hired crew member would not work. This is now fixed. Thanks ClownTastic for pointing this out.]