Nobody teaches this part, so people improvise: a mumble, a flipped king, a chair pushed back, and an opponent left staring at the board wondering whether the game is over or whether that was a move.
The thirty seconds it takes
Say it. “I resign” is enough, and Art. 5.1.2 gives the game to the player whose opponent declares that he resigns, ending it there and then.
Stop the clock.
Offer your hand. That one is custom rather than law, and it costs nothing at all.
Then fill in the result on both scoresheets and sign both, which is what Art. 8.7 asks for, and report the score the way your event wants it reported.
What you must not do is stand up and walk off. Your clock is still running until someone stops it, the arbiter has no idea what happened, and you need permission to leave the playing venue in the first place. Ten seconds of admin saves the arbiter a trip and saves you a result recorded as a loss by default.
Where the tipped king comes from
Films, mostly, and photographs of grandmasters. It is a nice piece of theatre and every experienced player understands it, but there is no article behind it.
At a junior event, use the words. An eight-year-old opponent may genuinely not know what a king lying on its side means, and I have watched a child play on for six moves against an empty chair because the resignation was silent.
When to resign, which is later than you think
At club level, being a piece down is not a lost game. It is an unpleasant game. The result comes from what happens in the next twenty moves, and plenty of that is still yours to influence.
Reasons to keep playing that come up constantly: your opponent has three minutes left and no increment; the ending is opposite coloured bishops, where a whole extra piece is often not enough; your king is exposed enough that a perpetual check exists; there is a stalemate trick in the position because you have almost nothing left.
The test I give students is a sentence. You may resign when you can say in one sentence what your opponent’s winning plan is, and you are confident they can carry it out. “I am a rook down and they will trade every piece” passes. “I feel terrible about move fourteen” does not.
Being lost is also a skill worth practising, because the moves that save half a point from a losing position are a specific kind of move: the check that repeats, the exchange that kills their last attacker, the pawn that walks. Defensive move puzzles drill exactly that, and they are the reason some players are hard to beat.
Then enter the game into an app the same evening and find the move where it actually went wrong, which is almost never the move you resigned on. Your own scoresheet is the only copy you will ever have.