If you're using a commercial inverter (and I assume you are), then I seriously doubt it will have any effect on your battery or alternator. These things should be designed with safety and reliability in mind, and so if there's any harmful effect on your vehicle, it will say so somewhere in the manual or packaging. If you think it's causing some damage and it doesn't make any not of this in the materials, just call up their support line and ask them.
My friend and I used to use power inverters in our cars to run home stereo equipment and we never had any problems.
If you have say a 500W power inverter it will be drawing around 42 amps of juice from your battery and alternator. Most alternators(even small cars) are at least 60 amps and many are 80 amp. your alternator and battery will be working harder than normal, but it is comparable to putting a large amplifier in to run subwoofers.
There are a few things you can do to help your situation:
1. don't let your car just idle for extended period- the alternator is most efficient around 1500-2000 engine RPM.
2. Buy an extra battery and mount in rear of car. they make isolators so the battery can power your stereo with the car off without draining the main battery and then is charged with the car running.
3. Replace your alternator with a high output model. Some cars you can get these from a parts store or there are many car audio companies that make high output alternators.
This is very helpful to keep your battery charged esp. if you have dual batteries.
Eventually I will have three batteries and two alternators on my car along with a capacitor system because I will be doing car audio competitions. I have one extra battery right now and it makes a great difference.
The rating on an inverter is the OUTPUT wattage it can sustain (they often specify peak ratings as well for short durations). Just because you are using an inverter that says 500W doesn't mean it's going to instantly draw 500W from your car battery no matter what load it has.
So don't go swap out your alternator or go adding additional car batteries yet! You won't see your carMP3 player drawing 42 amps from your alternator.
Take my inverter, for example. I have a LinkSys 140W inverter. It draws 80mA (0.08 Amps) at idle with no load on the output. Powering my entire mid-tower PC with several peripherals, it draws only 5.5 Amps (about 70W on the 12V DC side). Most vehicles can provide this much current with no problems. And I wouldn't worry about an inverter harming your vehicle's electical system -- the most that could happen is some 60Hz noise might get introduced because of a noisy inverter. Keep in mind that the inverters usually have a built-in fuse in their cigarette lighter plug and that your vehicle also has a fuse before the cigarette lighter socket, so you're pretty safe in terms of shorts, etc. if your inverter malfunctions.
Also, on PC switching PSUs, just because they say "230W" doesn't mean they're going to draw this much power no matter what and thus require a 250W inverter to run. Most carMP3 systems only draw 50-60W on average, so even a tiny power supply and cheap inverter will work just fine.
ChapAudio has excellent suggestions for beefing up your car's electrical system for high-end audio applications. Adding a high output alternator, extra batteries (especially high-current gel type batteries) and capacitors can help take the load off of your car and make your audio system cleaner and more powerful.
Just clearing up some misconceptions I've noticed on this and other boards...
I haven't gotten my mp3 player all together yet so I didn't know exactly how much of a draw it had.
With only 5.5 amps everything I suggested would be way overboard.
Thanks Jason, for clearing all this up.