DRM is digital rights management. Some call it digital restrictions management. Large companies like Microsoft, Apple etc. try and make you think its about protecting you or something. Basically, the music industry, film industry, etc., don't like having their stuff copied and having their copyright broken.
With CDs etc. there's not a great deal they can do about it. With DRM, until its cracked, it can stop it from being used in certain situations. For example, any music you buy from iTunes Music Store can only be used on 5 computers or devices at any one time I think -- if you buy a CD, you can use it on as many devices as you can. Some, including myself, think this should be your right. You have bought it, after all!
Many are against DRM because it goes against what rights you have; what freedoms you should have. Some just go against it because its a pain in the neck though. For example, right now only devices supporting Apple's DRM can play stuff with their DRM. This means iTunes and iPod. If you like other music software or other devices, then you're stuck. If Apple stop support for this DRM (they will do, eventually), you'll be stuck with using old software and devices to play it back with, unless you crack it. But cracking it is illegal anyway, so with that in mind you may well have just downloaded it illegally anyway.