MudPie says the common understanding. But there's more then this. Cog/ring teeth don't change their pitch as they wear, unlike the chain. They do change their shape and thus where in the "valley of the tooth" the chain seeks it's point of engagement. The more tooth wear there is the less the verticalness of the valley wall and the more the chain will want to ride up that wall, till it can no longer pull the tooth without slipping over it's top. This is the skip and grab we seek to avoid. As the chain wears and it's pitch does lengthen the more the last tooth on the cog will bare the forces we apply when pedaling. The more that tooth will wear at it's upper part of it's "wall".
This is why a partially worn cog can agree with a new chain, to a degree. But too much cog tooth wear and the tooth is more like a ramp then a wall. With too much pedaling pressure the chain will want to climb up that ramp shaped wall. Thus on the stand all seems well but in real life the chain skips over the cog with each pedal stroke. This point is the often misunderstood one. Cog teeth don't change their pitch, chains do.
Another aspect not well understood by some is that chain wear tends to be fairly linear referenced to miles (of similar conditions) but the tooth wear tends to be more geometrical with miles. So if , say, a chain and cog have a 2000 mile life and the chain is measured to have 75% of wear the cog will not have the same amount of wear. But in that last 25% of chain life the cog will see the rest of it's wear. Thus many chain checkers have a .75 and a 1 gage. Replace your chain before that 75% point and the cog will last for a few chains, extending it's life.
What some also miss is that chain performance is not solely restricted to cog skip as it wears. the chain's lateral flex increases too. This affects shifting, especially on the crankset. But this is much harder to measure and thus is not talked about much. With no number to post why bring it up
Why is because improved shifting is had with less worn chains. Since front shifting is more technique dependent this loss of performance dur to side to side flex increasing is, again, lost in the "shadows". Andy