How do players signal that they want to draw or resign?

Whenever I watch tournament footage, it seems like one player simply offers a handshake and then both walk away knowing what the result was (resignation or draw). Is there an accepted gesture or phrase to signal resignation or offer a draw? How does this work, especially between players that don't speak the same language?

Clearly the "signals" or "gestures" are agreed upon in advance either by the players or the more-likely - by the club/organization hosting the event. Usually though, a resignation is signaled by the player intentionally laying down their King horizontally on the square (knocking it over basically, but in a gentle manner that was clearly intentional); a draw is typically signaled by a handshake (hands shaking meaning draw accepted). 

However, this of course is susceptible to variants as there are often also in the room translators, moderators, officials and so on. Also, not all tournament footage is seen "live" where editing may also take place before any audience views it. 

This would be my limited insight, but I should also note that I have yet to ever take part in any international chess event where multiple languages are present.

I'm talking mostly about FIDE sanctioned events if that helps

People don't resign unless the position is terrible, so there doesn't need to be a signal. If you offer your hand / stop the clock it's obvious what's going on.

If a player stops the clock and then offers a handshake that is indicating resignation.  If a player offers a handshake with the clock running that is a draw offer.

To resign (which I have to do altogether too often), I lay my king on its side and offer a handshake while saying "Good game". I thought that was standard, but maybe it was just standard 50 years ago. 

To offer a draw, my understanding is that you make your move, say "Draw?" or words to that effect, and then hit the clock. If your opponent makes a move after that, it is a rejection of the draw offer. If he shakes your hand instead, it's clear that he is accepting a draw.

I've never seen anyone knock over the king on its side.

Taking the king between two fingers I'll tip it over, then set it back up on its square, and shake my opponent's hand.

At the pro level though, they just shake hands.

Also there's usually a fair amount of body language. In a difficult position if someone is hunched over the board the last few moves, then relaxes, lets out a sigh, and offers their hand, it's pretty clear they're giving up.

A resigning handshake isn't hard to tell from a drawing handshake. Let's say for an example, you fork your opponents queen and king in the end game, and your opponent shakes his head, lets out a sigh, and extends his hand. It's pretty clear he's resigning.

With a drawing handshake, I personally extend my hand and say draw in a questionable way. But you have to ask for a draw after making your move and before hitting the timer, as it says under the FIDE rules. I personally think there's no need to knock over your king and stuff, but maybe decades ago it was normal.