If your chain is worn so that it enlongates or "stretches" by more than 1%, more rapid wear of the cassette cogs and chainrings will result, as well as increased noise and slightly increased power loss. If wear continues well beyond 1%, at some point the chain will skip under heavy loads, ie, start jumping cogs. Putting a new chain on a worn cassette can also lead to skipping.
Rate of wear depends on a lot of factors, including lubrication, riding environment, and loads. Dirty or sandy roads and rain aren't good for chains at all. My experience has been that chains last about 5K miles before hitting the 0.75% wear point. I change the cassette every two chains, because I learned that putting the third chain on a cassette (after about 10K miles) results in the dreaded chain skip.
Chainrings wear too. They have more teeth to share the load, but of course are made of aluminum, not steel like the cogs. I've changed chainrings once on my current bike at 18K miles. Did this because I had excess "play" in a crankarm on the middle chainring, as well as noting that most of the chain wasn't being loaded when I pressed on the crankarm. New chainrings restored the "like new" smoothness and quiet under load.