The classic Cherry MX Blue switch is without a doubt the most famous clicky switch. You can still find it in many new mechanical keyboards today, but there are some more interesting and frankly better clicky switches if you're willing to dive into the custom keyboard community. Of all the clicky switches I've tried, the Novelkeys x Kailh Box Pale Blue is the best. This is a heavy switch with a sharp double-click thanks to Kailh's click bar design.
As a Box switch, it shares some basic housing features with switches like the Input Club Hakos and Novelkeys Royal. The square stem has a standard Cherry-style cross connector in the middle, so it'll work with most modern keycaps. The stem helps stabilize the plunger as it moves up and down inside the switch.
The "box" part of a Box switch doesn't refer to the stem but to the closed-off contacts inside. They're housed in a little closed off box with a plastic nub extending to touch the stem. You can see the green nub in the image below. The stem moves the nub, and that in turn pushes on the contacts to activate the switch. This is true of all Box switches, but the Pale Blue also has Kailh's innovative click bar design.
In Cherry's clicky switches, the "click" comes from a sliding jacket on the stem smacking into the bottom of the housing. The click bar is a small springy piece of metal that runs across the width of the switch housing, and it clicks on press and release instead of just on the press. There is a small bump on the side of the stem that has nothing to do with actuating the switch—it's entirely about making noise. As the stem goes down, it pushes the click bar down until it snaps back and hits the housing. Click. As you release the switch, the stem pushes the click bar out until it snaps back to strike the housing. Click again.