DRM stands for Digital Restrictions Management.
For a non-biased explaination of DRM, check out Wikipedia's Article.
Companies want your money, but don't want to give you anything. Its true. In a nutshell, DRM is about restricting what you can do with the things you buy. For example, if you buy music on Apple's on-line store, the DRM stops you playing it on a music player of your choice. They are trampling all over your Fair Use Rights.
When you buy a film on DVD, you should have the right to make a backup copy. You should be able to play it on any device you want (any DVD player, any PC etc). However DVDs have built in encryption (called CSS) to stop you doing this. Strictly speaking CSS isn't DRM, but the end result is the same - it takes away your Fair Use Rights. Fortunately CSS is completly useless, and was cracked by DVD Jon. The same guy has also cracked the DRM used by Apple's on-line music store. HD DVDs have also been cracked.
How does DRM work?
When you buy an encumbered DRM movie, you get two parts. The first part is a key, the second part is the movie which has been encrypted. On your computer (or inside your dvd player), there is a piece of software that knows how to decrypt the movie using the key. Without that software, the movie appears to be meaningless, random junk. Now here's the annoying part : the software that know how to decrypt the movie prevents you from doing what you want with your movie. For example, it won't make a backup copy, it won't put a copy on your computer, it won't let you watch it fully on many systems.
A Car Analogy
Imagine cars were protected using DRM. When you buy a DRM car, it comes complete with a key on a string, permanently tied to the wing mirror. Everybody can see the key, but the lock is cunningly hidden. You don't know where the lock is, so to use your car, you have to ask a robot to open the car for you. (The robot is an analogy for the piece of software or your dvd player). This robot will only open the car when it feels like it. If you do something it doesn't like, then it won't open the car. For example, Sony might sell you a car that will only be opened by the robot at night, restricting you to night time driving only.
People aren't dumb, and if they look closely at how the robot opens the door, they will be able to copy the robots actions, making the robot completely irrelevant. You would then be able to use your car whenever you wanted, rather than when the robot allowed you to. The car's DRM has been "cracked".
The US government passed a low making just looking at the robot illegal. In fact, even if you look at the robot outside of the US, then you can still be banged up in a US jail. Oh yes, the US truly is the land of the free - free to imprison whoever they like!
What Media isn't encumbered with DRM?
Most CDs don't have DRM. If you see the "Compact Disc" logo, then there is no DRM.
i-tunes has proposed to offer DRM free downloads for some of there songs, at a higher price.
What Media is encumbered with DRM?
Very few CDs are encumbered with DRM (most notably published by Sony). There is no way to be sure if a CD is damaged with DRM. Avoid Sony, and you should be ok.
Almost all DVDs have DRM. Fortunately, the DRM is so ineffectual, that circumventing it is trivial, but annoying. (Note : people where jailed to get us this far). In many countries it is illegal to distribute the software which lets you freely watch DVD movies, so it is never on a new computer (or a new operating system) out-of-the-box. You have to install it yourself.
Almost all HD-DVDs are encumbered. This DRM has also been cracked, but I don't know how annoying it is to jump through the hoops to circumvent the DRM.