Websites I like which haven't been submitted already:
http://www.homestarrunner.com - OK, so one person already MENTIONED it, but they didn't link to it, and all they said was "Shame on no one for mentioning it." So I'm mentioning it. Very wacky, very strange webtoon with lots of 80's references, in-jokes, and kooky humor. Most impressive thing about it: One man does every single voice in the cartoon, except for Marzipan (the girl).
http://www.lemondemon.com - Homepage of offbeat musician Neil Cicierega, best known for his song "The Ultimate Showdown of Ultimate Destiny."
Some of his songs do contain profanity and references to violence, but there are also plenty of silly, ordinary songs. Check the "Downloads" section. I recommend "This Hyper World," "Action Movie Hero Boy," and "Fine," which are some of his more "ordinary" songs. If you REALLY like wacky stuff, give "Mold en Mono" a listen--it's completely insane, but somehow quite enjoyable, especially since it almost, but doesn't quite, make sense. And if you scroll down to the "extras," there's a mad catchy remix of his song "Dance Like an Idioit."
http://www.creaturebreeder.com - An ultra-cute simulation game where you raise, breed, and sell different species of cute creatures. There are tons of variants on each species, and it's fun to try and breed for specific traits. It's also very easy to play. One of my favorite aspects of it is the fact that you get to decorate your creature farm with different plants and decorations. It's a bit like Animal Crossing in that respect. My only gripe is that it's somewhat slow to play--everything happens in real time, and creatures and eggs take a long time to mature/hatch, but otherwise it's quite well done. (My username is Freezair there, bee tww doubleyou.) And SPEAKING of creatures...
http://creatures.wikia.com/wiki/Creatures_Wiki_Homepage - The wiki for the Creatures series of games, a very complex, very intriguing, and totally addictive life simulation game. Follow the links on the front page to the "Docking Station" page and download Docking Station. (The DL link is on the bottom of the page.) Then check out the "C3/DS" section and download lots of Norn breeds, agents, rooms, and other fun stuff and kit out your space station will all kinds of fun junk. Then start raising Norns! They're funny little critters, and it's fun to just watch them interact with their enviroments. If you get really into it, you can read up on their fake genetics and learn about all the totally crazy stuff that goes on in this game's engine.
http://nifflas.ni2.se/ - Homepage of Nifflas, an independent game creator. Check out "Knytt" and "Knytt Stories." These are simple, atmospheric games that focus on exploration. They're very easy-going and very "pretty" to play. If you like more difficult games, check out "Within a Deep Forest": It's one of the most punishing, yet addictive, games you'll ever play. It's quite beautifully eerie.
http://www.jayisgames.com - A great resource for webgames. Links to lots of games, with reviews, ratings (Kid-Safe, G, PG, et cetra), and the like. My personal recommendations are "Sling," "Bloons," "Rapture Capture," and lots of stuff by Nitrome.
http://www.orisinal.com - Beautiful pastel Flash games; great for lunch breaks. Mostly arcade-style games.
http://eyezmaze.com/- If you like puzzles, then this is the site for you. No doubt you've played Grow by now. But have you tried Chronon, Galves' Adventure, or Grow Cube? Nope, thought not. So give it a try!