Finding out about this stuff changed my entire approach to getting things I need. Seriously. I no longer buy furniture. I often see appliances and if mine die I won't look to buy first. I've gotten clothes, shoes, toys, food---free.
There are sites online that link people who want to get rid of stuff to people who want stuff. Craigslist is the biggest national site, although it is more active in some places than others. Some of my friends swear by Freecycle. Family and friends in Utah use KSL's free stuff page. I'm sure there are other sites. The key is to find the one that is most active in your community and use it. Read it daily, and then jump when you see something you need.
Another source of free stuff in some communities is thrift stores that take donations on-site. They often receive stuff they don't or can't sell (like furniture that won't fit on the sales floor, toys when they don't sell toys, pots without lids or lids without pots, ugly art with nice frames, etc). Many of these stores (not including DI--and I can't figure that out!) can't afford the high dump fees to throw all this stuff away, so they put it out for free (often by the back door in a free section inside, or outside near the donations center, or sometimes out beside but not in the dumpster where you can take it until the store closes, when it goes into the dumpster). I do know people who actually dumpster dive thrift stores for the rejected but usable donations, but I don't recommend this because it's illegal in most areas.
In Las Vegas, a major source of free stuff (and recyclable metals for families that collect and sell them) is trash night. I know, it sounds terrible, but it's not. The trash collection in Vegas happens twice a week and isn't automated, so they take just about anything you can put on the curb. So people here put things that other people would truck off to a thrift store as a donation out on the curb for the trash man instead. And other people go wander around just after dark (it never actually gets dark here--it's always just after dusk, all night) on trash nights and collect the usable stuff that nobody wants. We've seen park benches, computer parts, toys, furniture, carpet, appliances, doors, windows--even a "marble" bust and its pedestal.
Another source of free stuff I've found is churches. Many churches have "recycle" or "swap" or "trading table" nights where anyone (not just members of the congregation) can come and bring what they want to get rid of and take what they need or want. It's a great way to get rid of and get baby clothes, housewares, small appliances, books, toys, etc. I was in a ward that did this every Enrichment night, and it was the only time I always went to (and didn't forget) Enrichment. The wealthy people in the ward eventually nixed it (and I stopped going to Enrichment) because they couldn't see any value in it (why would anyone want old used stuff?)--but I think every ward should do this, and then afterward take the leftovers and donate them to an appropriate charity (DI, Savers, the homeless shelter, etc).
If you know of active websites or locations for free stuff (NOT those stupid promotional sites that promise free laptops and are mostly scams, and not sites for free coupons or samples --that's a separate topic--but sites where you can get free real items from real people), post the links in the comments so the list here can become as complete as possible.