checking is a simple method of verifying that your autoloading pistol
does indeed have a round in the chamber and is ready for action
Why one would want to do this will baffle most people as it is more common to check and verify that the gun is unloaded rather than loaded.
Though it is true that it is best to store, transport, and carry firearms unloaded there are times when it could prove to be more dangerous to have an unloaded gun. For example, if you are out in the middle of the night and have your ccw piece with you which you think is loaded but really isn't because you forgot you unloaded it and didn't push check it before you left the house, should you ever need to draw and fire to protect yourself all you will hear is a click, that click will be the loudest sound you will ever here because you were expecting a bang.
To perform a push check all one needs to do is put the pistol on safe (if the design allows manipulation of the action while the safety is engaged other wise put the safety off) and with no finger touching the trigger gently push on the front of the slide until there is about 5mm of clearance between the chamber and the breech, you can now look inside and see if the gun is loaded or not. You don't want to push it far enough to extract and eject any round that might be in there, just far enough to have a look see.
While this might seem dangerous, it can be done safely with proper training, it is definitely safer than looking down the barrel and checking if there is a round in there.
Depending on your intention you can now proceed to loading or unloading the pistol as needed.
If it is loaded and you just wanted to verify that indeed it was be sure to verify that the gun goes back into full battery after you release pressure, otherwise pulling the trigger might not fire it because the disconnector was engaged.
Some pistols like the Walthers have a built in chamber loaded indicator from which one can determine if the chamber is loaded.
Accustomise yourself with pistol handling safety first before trying to learn this, any mistake on your part could lead to unwanted holes in your palms.