The standard way to measure chain wear is with a ruler or steel tape measure. This can be done without removing the chain from the bicycle. The normal technique is to measure a one-foot length, placing an inch mark of the ruler at the side of one rivet, then looking at the corresponding rivet 12 complete links away. On a new, unworn chain, this rivet will also line up exactly with an inch mark. With a worn chain, the rivet will be past the inch mark. [For accurate measurement, the chain should be held under some tension -- either on the bicycle, or hanging. -- John Allen]
This gives a direct measurement of the wear to the chain, and an indirect measurement of the wear to the sprockets:
If the rivet is less than 1/16" past the mark, all is well.
If the rivet is 1/16" past the mark, you should replace the chain, but the sprockets are probably undamaged.
If the rivet is 1/8" past the mark, you have left it too long, and the sprockets (at least the favorite ones) will be too badly worn. If you replace a chain at the 1/8" point, without replacing the sprockets, it may run OK and not skip, but the worn sprockets will cause the new chain to wear much faster than it should, until it catches up with the wear state of the sprockets.
If the rivet is past the 1/8" mark, a new chain will almost certainly skip on the worn sprockets, especially the smaller ones.