What if I am left-handed and the clock is on my right?

You press with your left hand, reaching across the board, because the rule is about the hand that moved and not about which side the clock sits on. If it bothers you, ask the arbiter before the round rather than after it starts. They decide where the clock goes, and many will move it if your opponent has no objection.

What happens if I press with the wrong hand?

The first time, nothing beyond your opponent or the arbiter mentioning it. Improper clock handling is penalised through the ladder in Art. 12.9, which starts with a warning, and the rung above that is two minutes added to your opponent's clock. Nobody is going to lose a game over one absent-minded press in round one.

Do I have to press the clock at all?

Yes, in any game with a clock. Your move is not completed until you press, and until then your own time is running, so forgetting costs you and nobody else. The single exception is a move that ends the game: checkmate ends it at once, and you do not need to press to make the mate count.

Which hand do you press the clock with?

Watch a strong player in a blitz game and count the hands. One picks up the knight, puts it on f3 and travels back across the board to hit the clock. The other one stays in a lap for the whole game. That is not a style choice.

The rule, and the reason for it

Art. 6.2.3 of the FIDE Laws: a player must press the clock with the same hand that made the move. There is no version of the game where this does not apply, and it matters far more in a three-minute game than in a four-hour one.

The reason is time. Two hands means moving and pressing in the same instant, and a tenth of a second a move over sixty moves is six seconds, which is a lot of blitz games. It also causes accidents. A hand already parked on the clock is a hand that swipes a bishop off f4 on the way past.

The same part of the Laws stops you keeping a finger on the button or hovering over it while your opponent thinks. Both are also plainly distracting, which puts them under Art. 11.5 as well, and there is a page here on what you may and may not do at the board.

Art. 6.2.4 asks you to handle the thing properly on top of that: no slamming it, no picking it up, no pressing it without having moved. That last one is not merely bad manners. Pressing the clock without making a move is treated as an illegal move under Art. 7.5.3, and the first illegal move costs you two minutes on your opponent’s clock.

Castling is where people get caught

Art. 4.1 says each move must be made with one hand only, and castling is one move. King to g1 with your right hand, rook from h1 to f1 with the same right hand, then press with it. Grab the king with one hand and the rook with the other and you have played an illegal move under Art. 7.5.4, with the two minutes to match. The order matters too, and there is a whole page on which piece you move first when castling.

Where this catches online players

There is nothing to press on a screen, which is the whole problem. The site takes your move and switches the clock for you, so a player whose first thousand games were online arrives at round one with no habit at all, moves with the right hand and presses with the left without ever noticing.

I did it in my own first round back over the board, twice, and a twelve-year-old put me right both times. Say sorry, put the free hand in your lap, and use the walk to the water cooler to build the habit: move, press, hands away, and then spend the time your opponent is using on checking what their last move attacked. That, rather than the clock, is where the games are actually lost.