Some folks don't know it's a thing.
Bicycle chains all have a 0.5" (12.7mm) pitch, which is the distance between the centres of their pins. It takes two sections of chain to make a link, meaning you have to adjust its length in 1" increments (except with single speed, where you might use a special half-link).
That's the theory, anyway... and a brand-new drivetrain is a sweet symphony, transmitting drive with 98% efficiency. But in practice, over time the pins and rollers become somewhat worn; the chain becomes effectively longer, and some of your effort starts to go into chewing up metal...
You don't really notice this happening, because it only happens slowly, and the lengthening chain deforms the chainrings and cogs over time, so it can be quite a while before any noticeable vibration develops... but your drivetrain is still losing efficiency all the while. Shift quality becomes affected, a throbbing vibration under power appears (it's the feeling of your chainring teeth being smooshed), and eventually the cogs or even chainrings can become so worn that the drivetrain poses a safety risk - the chain will jump over cog or ring teeth under high load, possibly dumping the rider on the road. It does take a lot of wear to cause this, except when worn parts are being mixed with new parts - a new chain on a worn cassette is the most common cause of dangerous slippage.
This wear is the result of an unusual combination; a worn chain on a new chainring:
...Not a happy marriage.
The chainrings, being larger, tend to wear more slowly than the cogs (cogsets come in cassettes or sometimes freewheels), which should usually be replaced with the chain. Some people buy two chains and rotate them every 1500km or so to get more life from an expensive cogset; others replace their chains when they're between 1/2 to 3/4 worn, before the chain has had a chance to cause much wear. Chainrings are more tolerant of wear; they can be used past the point it's visually obvious they're worn, but it's a bit of a grey area - they might look pretty good but not run smooth, or they might even run smooth but slip. To see if chainrings are still good, they're tested with a new chain and cassette to feel for vibration and slippage.
This one was obviously toast many km ago.
Cassettes are different - if you can see any wear, they've had it - this one is junk. It can be very difficult to say a cassette is good from a visual inspection; it might look virtually unworn but a new chain will jump over a couple of cogs... the vagaries of wear patterns can be subtle and confounding, which is why your LBS probably won't want to try a new chain on your old cassette unless you're changing it early.
So all this stuff sounds like bad news; this can get expensive with only medium-quality gear - ~$30 for a chain, ~$60 for a cassette, and around a dollar a tooth for chainrings... ouch. But then, aside from the occasional tyre, this is pretty much the only ongoing expense of maintaining a bike; it compares pretty favourably with a car on a per km basis. More good news is that if you're disappointed with the mileage you've got out of your gear before having to replace it, there's probably quite a bit of room for improvement if you're prepared to spend a little time looking after your chain.
Shiny rollers = dry chain
Another no-no is throwing lots of lube at the chain in the hope that more's better - it can be nearly as bad as not lubing, and can make a horrible, horrible mess... the excess lube attracts all manner of crud, which works its way inside the chain and becomes grinding paste. You only want the lube inside the chain, with no more than a residue elsewhere. Put a single drop of lube (I like @FBinNY 's Chain-L No.5) on each roller as you gradually backpedal the crank, letting the lube seep into the chain for a bit, before grabbing the chain with a rag between the rear derailer and the crank and backpedalling for a while. Lube just keeps coming off the chain, even when you lube it sparingly, and more will come off again if you go for a short ride. All this is worth it to purge the excess lube and keep the black filth at bay... the Nuke From Orbit option of taking the chain off and shaking it in a jar is somewhat fraught, IMO - are you washing more crud out of the chain than you're washing into it?
Nothing beats keeping it clean in the first place. New chains generally have the best lube; a light grease, it lasts for a long time before the first lube is required. A new chain will often have a sticky residue of this stuff all over it that will pick up dust like a magnet, so it's a good move to drag the chain through a rag damp with a little degreaser - after the chain's been through once, you feel it get suddenly slipperier through the rag. Often, this is all the cleaning necessary before a chain is to be lubed if it's been looked after (don't just aim a stream of degreaser at the chain), but a MTB or commuter can get pretty gungy in the line of duty... if it looks like it'll help, I hit the chain with a brush before I put any solvent near it.
Drivetrain maintenance is a bit of a fiddly mess (you should probably do it outside and wear gloves), but it's not too bad if you stay on top of it; it's rarely needed and easily done then.
Oil from the rollers, not the side plates