Word to the wise: If you're moving, no one wants your stuff. Don't leave it for the person who's taking your apartment, or try to give it to your friends. (Rob, this is directed at me, not you).
What do you do? If it's big stuff (furniture, electronics), post on http://www.craigslist.com and make it available for free. (If you're not familiar, craigslist is an online classified. And personal ads, but that's not important). If no one takes it, throw it out.
For smaller stuff, give it to Value Village/Goodwill, or throw it out. That's it.
Quote of the Day:
In the beginning the Universe was created. This has made a lot of people very angry and has been widely regarded as a bad move. - Douglas Adams, The Restaurant at the End of the Universe.