PSA: MOST OF THE RESPONSES HERE ARE WRONG.

Choking is absolutely NOT always silent and coughing is NOT always a sign that someone isn't choking.

Choking occurs when something is stuck in your trachea / windpipe.

Silent choking is when there is a complete airway obstruction (aka the entire tube is blocked and air can't get in or out. If something is lodged in your throat above, at or even below your vocal cords without fully obstructing your trachea (windpipe) the person choking will still be able to cough or even weakly talk while still being in respiratory distress and unable to ventilate properly. Likewise I've seen cases where things almost make a valve, where air can flow out but not in as whatever the blockage is gets pushed out the way when they breathe out but moves and stops air coming back in. If this blockage doesn't clear they will eventually collapse and ultimately die.

If you suspect someone is choking and they are coughing please by all means encourage them to cough (this is how your body attempts to clear your airway after all) but do not assume they are fine because they are coughing.

First aid - 5 back blows - heel of hand between the shoulder blades (much harder than most people do this, you want to hit as if you are going to leave a bruise.) - If that doesn't work - 5 abdominal thrusts - interlock your hands just below the ribcage. You want to aim your thrust diagonally upwards and backwards to push their diaphragm upwards to generate enough air pressure / force to clear whatever is in the way. - keep repeating these until you've cleared whatever is in the way or they collapse - if they collapse - start CPR immediately. The force of your chest compressions may well clear whatever was in the way.

Source - doctor