There have been many reports of discs getting scratched, or cracked from the Xbox 360 DVD drives.
MOST of those are due not to faulty consoles but user error, failure to read the product documentation, and failure to use common sense.
WHAT TO DO ON DAY ONE:
When you first get your console throw a disc in there that you don't care about. A Burned CD or DVD, if your console doesn't scratch this disc then you'll probably be good for the life of your console. Based on reports consoles don't just start scratching discs, they either always scratch them, on never scratch them (with the exception of user error). If you're worried your new console might scratch discs out of the box, test it out with some inconsequential backups of CDs or DVD movies (that's what backups are for).
If for some reason your console does scratch your test disc, without you having moved the console at all, then contact MS straight away to get a replacement console. Contact information for your area should be available in the included documentation, in the US they can be contacted at 1-800-4-MY-XBOX
DON'T MOVE YOUR CONSOLE WITH A DISC IN THE DRIVE
This is just plain, old fashioned, common sense. Your Xbox 360 is not a portable or mobile device, it's made for the living room, it was never designed with the intent to be moved. And even if you didn't have the smarts to realize your console should never be moved while a disc is in the drive it was pointed out in the documentation that came with the console (yes it was there on launch day). Apparently people still didn't get the message so MS has started including a warning sticker over the DVD tray that warns you not do move your console with a disc in the drive.
ESPECIALLY DON'T MOVE YOUR CONSOLE WHILE ITS ON
Moving your console while it's on is just plain stupid, again the Xbox 360 is not a portable or mobile device, it's made to be stationary. Not only will any discs in the drive get scratched to complete crap but you also risk damaging your DVD drive, and HDD (if you've got one). With the addition of wireless controllers there should be no reason your console would ever be moved while it's on, you can't even accidentally pull it with the controller cable anymore.
If you move your console while it's on you deserve whatever scratches your disc receives.
WHAT TO DO WHEN SHUTTING DOWN:
Never leave discs in the console, this goes beyond disc scratching issues. If you happen to move your console around your house a lot, or bring it to friends' houses, it's good practice to ALWAYS remove the disc before you shut the console down. If you get in that habit it will reduce your chances of leaving a disc in accidentally while moving it.
In addition to that, some games also have problems with creating corrupted game saves. This happens when auto-sign-in is enabled and the games boot directly from the disc. Both DOA4 and COD2 had issues with this (they have since been fixed with LIVE updates) but it's a good habit to not leave your disc in the the console when you're shutting down.
HOW TO HANDLE DISCS IN GENERAL:
-NEVER touch either flat surface of the disc, it's also good practice to not let anything else touch the flat surface of the disc either. Always handle it by the outer edge and inner ring of the disc, on an Xbox 360 disc these areas contain no data to reduce the risk of damage.
-When you remove your disc from the console it should go directly from your console into it's case, snap cases are designed to keep the disc safe, and they work much better then disc binders, or leaving your discs out in the open.
-If you absolutely MUST set the disc down, ALWAYS set it down on the "shiny side", NEVER set it down on the "label side". The way discs are designed the shiny side has a .6mm thick protective plastic layer, the label side of most discs offers no protection at all. If you scratch the shiny side it is probably repairable, if you scratch the label side, there is no way to repair the disc, you will have to buy a new one.
-If you happen to have dust or smudges/fingerprints on your disc you can remove these with a "Disc Wipe" these can usually be purchased at any store that sells games, DVDs, and/or CDs. Alternatively wipes made for eyeglasses may also be used. Be sure to only touch the surface of the disc with the wipe and not with your fingers or anything else. When wiping the disc it is recommended you start at the center and wipe out toward the outer edge, never wipe in a circular fashion. Wiping from the inner circle out will remove the dust and oils out and off the disc, while wiping in a circular motion will only spread it around and could potentially cause scratches.
WHAT TO DO IF YOUR DISC DOES GET SCRATCHED:
If your Xbox 360 console scratches a disc while following the advice above there are things you can do.
First test out the console with some backups of CDs or DVD movies to see if the console scratches those discs as well, if it scratches your test disc, it's probably not just a one time occurrence and you should contact MS for a replacement console.
If your console has scratched an MS Game Studios brand disc contact MS and request a replacement disc. MS can only replace MS discs. If a different game, DVD or CD is scratched, contact the publisher for that disc. Be forewarned that most will not replace your disc. Most that will replace it will also require you send back your broken disc.
If you are unable to have a new disc sent to you look into a disc resurfacer. The "Spin Doctor" is probably the most well known of these devices. Basically they strip off a very thin layer of that .6mm protective plastic (for example it might reduce it to: .55mm) If the layer removed by the resurfacer is thicker then your scratches are deep then you should end up with a smooth, unscratched surface. Because it removes plastic from the disc, it can only be used on a single disc two or 3 times before it has removed too much of the disc for it to remain readable. If you can't afford to buy a disc resurfacer there are services that will resurface your discs for you. This technique should only be used as a last resort.
VERTICAL OR HORIZONTAL?
Which position is less likely to scratch your discs? So far there is no data to suggest the one way or another is better or worse in terms of scratching discs in an Xbox 360. OTHER devices however (such as PS2s, PC drives, etc.) are less prone to scratches while in the horizontal position. If you have a choice I would also recommend the horizontal position over vertical due to the simple fact that the console is more stable in that position, and less likely to topple, or wobble from vibrations (be it from the 360's drives or some other source). The console should function fine in either position though.
BAD ADVICE THAT FLOATS AROUND THE FORUMS:
Keeping your discs scratch free is really a simple process, and it boggles the mind how some simply can't follow common sense. Even still some promote using OTHER methods of disc protection instead of just not being stupid. Here are some things you'll hear people recommend that are just flat out bad advice.
D-Skins are probably one of the worst things you can do to your DVD drive. While they DO protect your discs from scratches, the cure in this case is worse then the disease. Because they're so cheaply made they're typically not very balanced, this causes added vibrations that will not only make your drive more noisy but it will also significantly shorten the lifespan of the servo motor through having to push the extra weight (it DOES make a differences to such delicate electronics). In addition to motor burnout you also risk bearing run-out, which will probably occur before the servo motor burns out, even worse if bearing run out occurs it will make your drive far more likely to scratch discs you put in it. Both of these things can also significantly increase the noise level of your drive.
Past motor issues it also wreaks havoc on the laser. The added vibrations and added thickness of the plastic between the laser and the data, not to mention microscopic dust and other particles that get caught between the skin and the disk (which creates an un-even surface) cause the laser to increase it's power (which can cut it's lifespan in half) as well as risk burning out the focus servo on the laser array. In addition to that you increase the chance of miss reads which can lengthen your load times.
Foam Bumpers and other drive modifications
This is another bad idea. If your drive is scratching discs when treated properly, GET IT REPLACED, MS will replace your console without hassle, DO NOT TRY TO FIX IT YOURSELF. If you open your console you will void your warranty and MS will not fix your console for you. Past that, these "fixes" only supposedly protect you from the stupidity of moving your console with a disc in the drive, which you shouldn't be doing anyway.
This should cover everything you need to know about how to not scratch your discs, if I missed something PM me and let me know and I'll consider adding it.
Any topics that pop up about discs scratching will most likely be closed with a link referring to this topic, so save yourself the effort and don't bother.