I'm not trying to emulate the big applets collections here, but rather concentrate on a few professional-quality examples that look useful for applications in a business context. (Look elsewhere for the fun stuff, there's lots of it to be found in the collections listed below!) Some of the applets come with source code, others are commercial but sometimes there's a trial version available or an on-line demo.

Class libraries can now be found on the Java Development Supporting Tools page.

There's even a link here to my own first Java exercises: an on-line demo and downloadable classes and source code!

From Javasoft:

Applets from Javasoft For several there is now a 1.1 version available!
Imagemap Example from Javasoft

From other sources (alphabetically sorted):

Cafe del Sol
CButton Mark Qian - download link through PC World Software Library
Chris Cobb's Obligatory Date Applet
Dave's Java Gifts
FindItEasy demo - but free for limited use (DoubleOLogic Software)
HomePageSearch and other applets
HTMLViewer demo
JavaWorld - Java Step by Step Site navigator
Jlpr, the PostScript Applet by Ernest Friedman-Hill; warning: for unknown reasons this applet does not (yet) work with MS Internet Explorer
SE Java Applets
xProcs for Java Fabula On-line

Example Applets using the StringDateTime class ... my own! And free as well.

And of course, these are sorted as well:

Bill's Place
Digital Cats' Java Resource Center -English- Applets collection
Java Applets at the Netscape site
Java^TM Boutique, The A Collection of Java Applets
Nizze's Java Applets
PartBank - your Java component software source (not all free)
Software Library (PC World) Java, Plug-ins & Active-X
Welcome to JARS Ratings of Java Applets (not all free)

Again, alphabetically sorted by description, from various vendors:

Argent Software Coffee Table Applet 2.0 (Class version on the Java Development Supporting Tools page)
dbProbe 3.0 (InterNetivity Inc.)
DoubleOLogic Software
Datavista Pro Visualize, Inc.
eSuite Home Page Lotus eSuite, formerly codenamed Kona: a collection of light-weight office applications

No, not applets: applications! Some awesome, some not yet out of beta, but all interesting:

Ding! 2.0 A people browser from Activerse
HotJava^TM Javasoft's Java Browser
J Street Mailer
WallStreetWeb by BulletProof Java Investor application: register as Guest and poke around!
IBM Network Printer Manager
Moneydance Version 0.9.4 Keep track of your finances (Freeware!)
NetResults 1.2 Java-based, site-specific search engine
VisualRoute A traceroute application with a difference: not only does it give more information than Windows' own tracert.exe gives you, it also shows the route taken graphically on a map. (32-bit Windows only.)

Really important link missing? Broken link? Site moved? Other comments? Please let me know!