Two clocks in one box with a rocker between them. Press one side down and the other pops up, and only the raised side is counting. That is the whole machine, and the rest is procedure.
Pressing is part of the move
The order is fixed: make the move on the board, then press. Art. 6.2.1 says pressing your clock is what completes the move, and until you press, your own time is running. Those seconds between putting the piece down and hitting the button are yours, not your opponent’s, which Art. 6.2.2 spells out because people used to argue about it.
Two more rules live in the same article and both get broken by beginners every round. Press with the same hand that moved the piece (Art. 6.2.3), which also means no hovering a finger over the button while you think. And handle the thing properly (Art. 6.2.4): no slamming it, no picking it up, no pressing before you have moved. Improper handling is a penalty under Art. 12.9, and at junior events arbiters give those out. There is a separate page on which hand presses the clock because the one-hand rule surprises people.
Analogue, digital, and where the word flag comes from
An analogue clock is a wind-up dial with a thin metal flag hanging over the twelve. As the minute hand comes round it lifts the flag, and when the hand reaches twelve the flag drops. That drop is the flag fall, and it is why players still say flag about a digital display that has no flag in it at all.
A digital clock shows minutes and seconds, counts your moves, and can add time after each move or hold your clock still for a few seconds before it starts. An analogue clock can do none of that, which is why serious chess moved over: a control like 90 minutes plus 30 seconds a move is simply not possible on a dial.
Where it sits and who is allowed to touch it
Art. 6.5 gives the arbiter the say on which side of the board the clock goes. In most halls it ends up on Black’s side, and if you have a strong preference, ask before the round rather than after.
White’s clock is started at the announced start of the round whether White is in the chair or not. If a game has to be interrupted the arbiter stops the clock. You may stop it yourself only in order to call the arbiter over, which is exactly what you do when you are making a claim, and then your hand goes up and you wait.
The numbers on the display only mean something once you can read the time control, so that is the next page. And if you are the sort of player who freezes at two minutes, the cure is not clock practice, it is pattern practice: people who see tactics instantly never get into the scramble in the first place.