Chain stretch happens when the chain line gets longer due to use, and this is a natural occurrence. Actually the chain doesn't stretch like a rubber band, but it lengthens when the holes holding the chain pins grind down. A new chain will mesh nicely with your chainrings' and cogset's teeth, but when it stretches, it will eat away those very same teeth swiftly. Since chains are cheaper than chainrings or cogsets, it makes sense to change your chain before it begins eating the other parts. There are three ways on how to check for chain wear:
1) The Pull Method. (Picture 1) This is a fast and easy check. Put your chain on the big chainring and smallest cog gear, and then at the chainring area try to pull off the chain away from the crank. If you can see a lot of space around the teeth (as in the photo), your chain is worn. This method is not as accurate as the others but it is a good starting point.
2) The Measuring Method. (Picture 2) Count off 24 links (12 outer plates and 12 inner ones) and measure from the center of the pin at one end to the center of the pin at the other end. A new chain will measure exactly 12 inches. If it is 1/16 inch , it is time to change your chain soon. If it is 1/8 inch over, replace it now. If it is 3/16 or over, you should have replaced your chain yesterday!
3) Using a Chain Checker Tool. (Picture 3) There are many designs of this tool, and I have the Nashbar model. You simply drop in the tool and see if the "T" part sinks into the space in the link. If it does not, your chain is still OK. But if it sinks into the gap, then it is time for a chain change. Most checkers have indicators for 0.5% and 1.0% elongation which translates to 1/16 and 1/8 inch over 12 inches, both of which cases you need to change your chain.
I hope this helps! Next topic is how to change your chain.