Is there a button that pilots push to transmit on the radio? Is there any button which needs to be pressed when the pilot or co-pilot wishes to talk to ATC? 

Normally in videos on YouTube they don't seem to be doing anything special other than speaking, but I believe that there's more to it. Probably the push buttons (like on a walkie-talkie) are located very close to their hands?

I'm generally asking about all aircraft that communicate with ATC, not any specific airliner.

Please note that I know the handshake stuff and the frequencies and the radio part of it. My question is only about whether they do or they don't have a push button when they have to send a message. It doesn't seem to be an always on thing because obviously that would clutter up the channel and ATC guys don't need to listen to what the pilots are talking in the cockpit routinely.
There is a push to talk (PTT) button located at the yoke or sidestick, in small and large aircraft likewise. When the PTT button is not triggered, all communication is routed through the intercom and not broadcasted on the frequency. Once the button is triggered, communication is broadcasted on the active frequency.

In the below picture of a glider, it is located on top of the middle stick.
On the second picture is it behind the yoke, to be used with the index finger.

(Image Source: WikiPedia)


(Image Source: Private Blog)