There's certainly no shortage of downloadable applets on the Web - a Yahoo search on 'Java Applets' returned 328 sites, and that's just the tip of the iceberg. Here's a small selection of applet-rich sites to get you started.
Café Java Shoppe offers a wide selection of menu-builders, image-warpers, text-decorators, animated buttons and the like. There are no online demos, and the source code is often unavailable, but that won't worry you if youre not into Java programming. At the more serious end of the market EarthWeb's JARS.com is primarily aimed at developers, but check out its top 1% and 5% applets listings for some good stuff.
The UK's top .class site is the Java Centre (http://www.java.co.uk/javacentre.html), with a great selection of applets ranging from menu builders and ad banner to Space Invaders and a Sinclair Spectrum simulator - watch out if you;re using IE4, however, as I've had difficulties opening the individual applet pages due to a frame problem. The Java Boutique (http://javaboutique.internet.com/) has a similarly large selection (including a crossword puzzle), all with online demonstrations. Probably the biggest applet library of them all, however, is Gamelan (http://www.developer.com/directories/pages/dir.java.html), which includes 1092 items in the Educational category alone.
Finally, don't forget the King of the Lake, David Griffiths, whose site contains a small, but perfectly formed set of applets for you to download and use.